Research ArticleSynchronous Acquisition and Analysis of Ultrasonic SpectralInformation for the Characterization of Particle Size Distribution

Nan Jia,1 Jianfei Gu,2 Huinan Yang,1 and Mingxu Su 1

1Key Laboratory of Shanghai for Power Engineering Multi-Phase and Heat Transfer, University of Shanghai for Scienceand Technology, Shanghai 200093, China2School of Electrical and Automatic Engineering, Changshu Institute of Technology, Changshu 215500, China

Correspondence should be addressed to Mingxu Su; sumx@usst.edu.cn

Received 2 December 2018; Revised 27 February 2019; Accepted 12 March 2019; Published 2 May 2019

Academic Editor: Andrea Cusano

Copyright © 2019 Nan Jia et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Particle sizing methods have gained considerable attention in the past few decades, but there is still a big challenge in highconcentration situations (i.e., volume fraction > 10%). However, the ultrasonic spectroscopy technique is a common tool for thenoninvasive determination of essential parameters for high concentration systems by analyzing ultrasonic spectra with inversionalgorithms, including the particle size distribution (PSD), volume fraction of each phase, and physicochemical properties. Forthe ultrasonic measurements, proper acquisition and analysis of ultrasonic spectra are becoming significant in order tounderstand the relationship between the unknown parameters and the ultrasonic spectra. In the work, an experimental setupwas provided to synchronously acquire ultrasonic reflection and transmission signals. A series of experiments were performedon silicon-water solutions at volume fractions 8%, 10%, and 12% to obtain the ultrasonic attenuation spectra and ultrasonicphase velocity spectra based on different measurement methods, i.e., the pulse-echo method, reference reflection method, andthrough-transmission method, respectively. Based on the Epstein-Carhart-Allegra-Hawley (ECAH) forward model, geneticalgorithm (GA) and optimum regularization technique (ORT) algorithms were implemented to determine PSD with themeasured spectra; the obtained PSD was then compared with the optical microscope method. It revealed that the spectraobtained by different measurement methods showed individual features while the obtained PSD was consistent and the volumemedian diameters were within a deviation of 10% with GA and ORT algorithms. The differences and characteristics of thesethree measurement methods for signal acquisition and interpretation were discussed and presented to provide an evaluation andrecommendation for ultrasonic particle sizing.

1. Introduction

With the rapid development of the industrial processes forchemical, material, and pharmaceutical productions and foodprocessing, particle sizing technology has obtained consider-able concerns and investigations [1, 2]. In various dispersedsystems, the particle size obviously influences the materialproperties such as stability, appearance, and performance.For instance, the tinctorial power is usually dominated bythe pigment size, the additive size significantly affects the filmstrength and wear resistance, and the catalyst size and its dis-tribution also influence the catalytic activity [3, 4]. Therefore,it is useful and meaningful to measure particle size in variousindustrial fields. The most widely employed measurement

techniques for particle size distribution (PSD) are based onstatic or dynamic light scattering and image analysis [5].However, compared with these competitive approaches, themethod of ultrasonic spectroscopy also possesses the remark-able advantages in the highly concentrated suspension andeven in the optical opaque medium, due to the strongpenetrability of ultrasonic wave propagating through themedium [6–9]. Meanwhile, this approach can also permitthe noninvasive and nondestructive measurements to avoidcontaminating samples as well as allow measurements undercircumstances close to process conditions [10, 11]. Further-more, this method is also characterized by commerciallyavailable and endurable experimental setups, so it is rightthese characteristics that promote the ultrasonic spectroscopy

HindawiJournal of SensorsVolume 2019, Article ID 8251829, 12 pageshttps://doi.org/10.1155/2019/8251829

technique development and application, for example, mea-suring the concentration and size of the pulverized coal sys-tem [12] and monitoring the characteristics of the dynamiccrystallization process [13] and the process tomography forgas-solid two-phase flow characterization [14].

It is a truth universally acknowledged that ultrasound isan important carrier to reflect the size and distribution ofparticles in suspensions. The overall process for particle sizedistribution of a suspension measured using ultrasonicspectra can be focused on two aspects: one is the adequatemeasurement of frequency-dependent ultrasonic spectra,and the other is the appropriate implementation using theultrasonic theoretical model and inversion algorithm [15,16]. It is essential to carry out both steps carefully for anaccurate measurement.

On one hand, as far as the measurement itself is con-cerned, it is crucial to take extreme care in guaranteeing thedevice well-designed (e.g., a collimated beam and a smoothinterface of buffer rods) in order to acquire the reliable ultra-sonic spectra. In principle, it allows for in situ process mea-surements, given that ultrasonic wave information can beobtained in a fast and nondestructive way. The availableapproaches for particle size distribution based on the ultra-sonic spectrum technique are mainly classified into the ultra-sonic attenuation method and ultrasonic phase velocitymethod [17, 18]. And both spectra are measured after anultrasonic wave transmitting through the suspension withdispersed particles and can provide the similar informationsuch as the particle size and its distribution. But the sensitiv-ity and information contents of experimental data are differ-ent since the attenuation spectrum is related to the ultrasonicamplitude while the phase velocity is related to the phase shiftand time difference. Furthermore, different spectral analysismethods have a very good complementarity if their charac-teristics can be fully analyzed and utilized; for example, tem-perature is particularly essential for accurate phase velocitybut is generally less critical where the ultrasonic attenuationspectrum is concerned.

On the other hand, the theoretical model of ultrasonicwave propagation should be provided for the inversion calcu-lation, and the model contains the information of materialproperties, measurement conditions, and measured ultra-sonic spectra; the PSD can be solved with a mathematicalequation. By employing a rigorous and fundamental theoret-ical model, particle sizing with ultrasonic spectra deeplyrelies on the assumption that the fundamental relationshipbetween ultrasound spectra and requisite physical properties,and ultimately the inversion calculation of the resultant par-ticle size distribution. The accuracy of the retrieval resultswould be influenced not only by ultrasonic spectra butalso by the selection of inversion algorithms. Several inver-sion algorithms have been investigated and developed inthe previous researches to provide an appropriate way toanalyze the ultrasonic spectrum and obtain the PSD, forexample, the Davidon-Fletcher-Powell (DFP) algorithm,Twomey’s method and its improved form [19, 20], the opti-mum regularization technique (ORT) [21], Fruit Fly Optimi-zation (FOA) [22], and the genetic algorithm (GA) [23].Among them, the ORT is improved by optimizing the

regularization factors and can be used in solving the ill-conditioned equations with a higher stability of solution,and the GA is characterized by its university, robustness,and feasibility for parallel computing. Accordingly, thesetwo inversion algorithms are discussed to determine theeffective spectral information.

In the previous investigations, the particle size distribu-tion in suspensions was usually obtained by utilizing thesingle-mode ultrasonic method extracted from eitherreflection or transmission signals, which would causeuncertainty to the measurement results due to insufficientinformation. Starting from analyzing ultrasonic attenuationand phase velocity spectra and comparing the inversedparticle size and distribution, a comprehensive measure-ment method and a carefully designed experimental setupwere proposed to provide a noninvasive way in synchro-nously acquiring these two kinds of ultrasound signals inthe present work, which can effectively suppress the inter-ference of different ultrasonic analysis methods in synchro-nous measurement. In addition, the above two broadbandultrasonic spectra can be synchronously extracted fromsimultaneous experimental signals, which were implementedby the pulse-echo method, reference reflection method, andthrough-transmissionmethods at the same time. Particularly,the relatively novel method of the ultrasonic phase velocityspectrum was also used to inverse PSD with the ORT andGA algorithms. Thus, characteristics of the above variousmeasurement methods were discussed so as to facilitate asuitable understanding on the information content exploita-tion for characterizing particle size distribution in suspen-sions using ultrasonic spectra.

2. Measurement and Analysis Methods ofUltrasonic Spectra

2.1. Measurement Principles. Figure 1 shows the schematicdiagram of the acoustic propagation path with simultaneousmeasurement of ultrasonic reflection and transmission sig-nals. The measurement apparatus has a geometricallysymmetrical structure and consists of two pieces of plexiglassbuffer rods (delay line) with the thickness of l1, and measure-ment cell is l2. A pair of ultrasonic transducers were mountedon each end of the buffer rods; one can serve as transmitter/-receiver (transducer A); the other is the receiver with thesame parameters (transducer B). In the test run, the compres-sional wave A0 is first generated by ultrasonic transducer A,and it travels along the delay line/sample interface where itis partly reflected and partly transmitted. And then, thereflected wave A1 travels back through the delay line and isreceived by the same transducer. The transmitted portioncontinues to propagate through the measurement cell andis reflected both on the sample/delay line interface and thedelay line/transducer interface; for the same reason, thereflected waves also transmit back and are detected by trans-ducer A, which are termed as A2 and A3. Finally, transmittedwave A∗ is received by transducer B. The ultrasonic spectracan be obtained by measuring and analyzing the aforemen-tioned sound signals, and the ultrasonic attenuation coeffi-cient is a measure of the decrease in amplitudes of the

2 Journal of Sensors

ultrasonic waves (reflection or transmission) per unit dis-tance as they travel through the materials; the velocity isdetermined by measuring the elapsed time difference andtravel distance between any two different signals, whichwill be discussed in more details below. There is no con-sideration about the multiple reflections here due to thesharply decaying signals, and a low signal-noise ratio(SNR) often brings a great trouble in separating the effec-tive signals.

2.2. Ultrasonic Attenuation Analysis. In principle, the attenu-ation coefficient can be calculated by comparing the reduc-tion in the amplitude of the wave traveled across thesample with that of a wave traveled across the buffer rodor a calibration material. The measurement techniquesmay be operated in a pulse-echo method, reference reflec-tion method, and through-transmission method accordingto their respective principles.

2.2.1. Pulse-Echo Method. The pulse-echo method [24] onlyneeds one transducer to work; that is, a single transducer isemployed to both transmit and receive the ultrasonic signals.This method can provide an absolute measurement by ana-lyzing the sample echoes A1 and A2, while the latter transmitsthrough the measurement cell and echo A1 merely spreadswithin the buffer rod. The ultrasonic amplitudes areobtained by fast Fourier transform (FFT) with the time-domain signals as

A1 = A0Rse−2αc l1 ,

A2 = A0Rs 1 − R2s e−2αc l1e−2αs l2 ,

1

where Rs is the reflection coefficient of the delay line/-sample interface and αc and αs represent the attenuationcoefficients of the buffer rod and the sample material,with buffer rod length l1 and the measurement cell lengthl2, respectively. The ultrasonic attenuation coefficient can

be expressed as

αs =12ln A1 1 − R2

s /A2l2

2

It is difficult to determine A1 and A2 directly, butthey can be calculated indirectly by using air as the cali-bration reference, since the total reflection may occur inthe delay line/air interface due to the very small acousticimpedance of air compared to that of the buffer rod (e.g.,perspex). A group of reflection signals Aa can be mea-sured for the vacant measurement cell, as expressed asfollows:

Aa = A0e−2αcl1 3

The reflection coefficient of the delay line/sampleinterface Rs can be represented by the following equation:

Rs =A1Aa

4

It can be seen from equation (4) that the reflectioncoefficient of the sample is determined by measuringthe amplitudes of the pulse echoes reflected from theend of the first buffer rod when the sample and the airare present, respectively. So, the ultrasonic attenuationcoefficient of the sample can be calculated as

αs =12ln A1 1 − A1/Aa

2 /A2l2

5

Owing to Rs being expressed as the quotient of twopulse amplitudes of A1 and Aa, the problem of measuringabsolute signal amplitudes A0 may be avoided, which notonly simplifies the expression of the ultrasonic attenua-tion coefficient but also improves the measurementaccuracy.

2.2.2. Reference Reflection Method. The reference reflectionmethod [25, 26] belongs to relative measurement, and it uti-lizes the reflection wave A2w for the reference water andreflection wave A2 for the sample; they both spread throughthe measurement cell but possess different amplitudes onaccount of encountering different materials. This workingmodel is also subjected to pulse-echo measurement with anexpression of the ultrasonic attenuation coefficient as

αs = αw + 12ln A2w A1/Aa 1 − A1/Aa

2 /A2Rw 1 − Rw2

l2,

6

where Rw is the reflection coefficient of the delay line/waterinterface and αw is the water attenuation coefficient.

It should be noted that the reflection signal A3 that trans-mits through the measurement cell and is then reflected onthe delay line/transducer interface is not utilized here,

Transducer A Transducer BSample

Buffer block

l1 l1l2

A⁎A1A0A2A3

Figure 1: Schematic representation of ultrasonic propagation in themeasurement cell.

3Journal of Sensors

because of the interference effects and the sharply attenuatedecho signals.

2.2.3. Through-Transmission Method. The through-transmission method [24] is also a relative measurement inwhich water transmission wave A1w and sample transmissionwave A∗ are adopted. The working model belongs to thepitch-catch principle with two separated transducersemployed to transmit and receive thewave signals. Theexpres-sion of the ultrasonic attenuation coefficient is

αs = αw −ln A∗ 1 − Rw

2 /A1w 1 − A1/Aa2

l27

It isworthnoting that the above threemethods are suitablefor single-frequency ultrasound processing and can also beusedforultrasonicspectrumanalysis. Inthiscase, thespectrumneeds tobeobtained in advanceusing the aforementionedFFTtechnique, and themultifrequency components shouldbe cor-respondingly calculatedwith the amplitude at the correspond-ing frequency, whereby the respective acoustic attenuationspectra can be obtained from the amplitude spectrumanalysis.

2.3. Ultrasonic Velocity Analysis. As mentioned earlier, thevelocity spectrum is only related to the phase and hasnothing to do with the amplitude, so it mainly dependson the time-of-flight (TOF) and the phase difference. Sincethe pulse-echo method adopts the same testing material, itcan hardly be used for the calculation of phase velocity.The through-transmission method is available in principle,but the reference reflection method experiences a soundlength twice as far as the former, consequently producinga more distinct phase difference; hence, it is preferablefor the reference reflection method in calculating ultra-sonic velocity.

When the measurement cell is full of water (as a refer-ence), the time that the ultrasonic wave transmits from thetransducer A and then is reflected is recorded ast1, whenthe measurement cell is full of sample, for the same propaga-tion process, the corresponding time is represented as t2. Thepropagation time can be treated as

t1 =2l1cc

+ 2l2cw

,

t2 =2l1cc

+ 2l2cs

,8

where cc, cw, and cs are the ultrasonic velocities of the bufferrod, water, and sample, respectively (cw is known accurately).

It is crucial to firstly calculate propagation time difference(t1 − t2) to solve cs. The formulas are as follows:

cs =cw

1 − cw t1 − t2 /l2,

t1 − t2 =Δpsω

,9

where Δps refers to the phase shifts of the two reflection sig-nals and ω is the angular frequency.

Since the ultrasonic spectra are fundamental physicalcharacteristics and highly rely on the composition andmicro-structure of thematerial, they are closely related to particle sizeand distribution. Thus, it is important to keep the measure-ment device in good condition to obtain reliable and accuratedata. Equally important is the way analysismode inwhich dif-ferentmeasurementmethodsserve toobtain thespectral infor-mation components and correlations.

3. Experiments

3.1. Sample and Experimental Setup. The silicon particle sam-ples (as shown in Figure 2) were employed in the experi-ments, which could be treated roughly as quasisphericalparticles. Figure 3 illustrates the particle size distribution ofsilicon samples measured by the optical image analyzer(model Pip8.1, Omec Inc.), and it denotes a relatively narrowdistribution from 7.56 to 15.24μm with the volume mediandiameter of 11.37μm of the particles. In order to avoid thesedimentation of samples, the experiments should be carriedout once after preparing the three silicon-water solutions atvolume fractions 8%, 10%, and 12%.

The measurement setup configuration and its workingprocess are demonstrated in Figure 4. According to the sizerange of silicon particles, a pair of ultrasonic longitudinalwave transducers (Panametrics V313-SU, nominal centerfrequency 5MHz) are employed and aligned on both sidesof the detection area. Transducer A serves as the transmit-ter/receiver transducer, working with the ultrasonic pulsesignal transmitter/receiver (Panametrics 5800PR, highestexcitation frequency 35MHz) in the pulse-echo manner,and transducer B is the same model but only takes effectin receiving the ultrasonic signals combined with anotherultrasonic pulse signal transmitter/receiver (Panametrics5077PR, highest excitation frequency 10MHz). The mea-surement apparatus has a geometrically symmetrical struc-ture and consists of two pieces of plexiglass buffer rods withthe thickness 10mm and measurement cell 6.5mm, wherethe buffer rod should be optimized carefully in advance inorder to avoid possible confusion due to an echo interfer-ence phenomenon. And there are three factors to be con-sidered when selecting appropriately for the optimalexperimental results: smoothness, thickness, and material.Firstly, the surface of the buffer rod must have a goodsmoothness to guarantee that the ultrasonic transducercan be closely aligned with its surface; secondly, the thick-ness of the buffer rod should separate the reflection signalseffectively; and finally, the buffer rod should possess a suit-able acoustic impedance to ensure a good signal-to-noiseratio (SNR). One more concern is that the temperatureshould be carefully kept around 25°C by adopting a ther-mostat bath to avoid the effects of temperature changeson concerning sound propagation. To acquire the data ofreflection and transmission signals and analyze the wave-form synchronously, a computer with a dual-channel dataacquisition card (NI PCI-5114, with a sampling rate of250MS/s) was employed.

4 Journal of Sensors

3.2. Experimental Phenomena and Signal Analysis. Duringtesting, the signals were acquired after adjusting the ultra-sonic spectrometers: the ultrasonic reflection signals wereunder the condition of the gain of 40 dB and transmittingpower of 50μJ (5800PR), and the transmission signals wereunder the circumstance of the gain of 20 dB (5077PR). Theexperimental reflection and transmission signals of silicon-water solution with the volume fraction 12% are shown inFigure 5. A0 is the initial signal, and A1, A2, and A3 are thereflection signals of the delay line/sample interface, sample/-delay line interface, and delay line/transducer interface,respectively. A∗ is the transmission signal ultimately reachingthe transducer on the other side. It can be seen that A1 expe-riences the shortest distance, A2 spreads two more measure-ment cell length than A1, and A

∗ only goes through one moremeasurement cell length than A1, so it is obvious that thephase of A∗ is behind A1 but ahead of A2. The reflection sig-nal A3 travels the longest distance and has the weaker ampli-tude compared with other signals, which as mentionedearlier, and is not used here.

The reflection and transmission signals can be applied toanalyze the ultrasonic spectra through spectrum analysis

based on the fast Fourier transform. The ultrasonic attenua-tion spectrum related to the signal amplitude can be obtainedby using the reflection signals such as A1 and A2 or the refer-ence signals in the samecondition. Similarly, thephasevelocityis associated with the unwrapped phase shift, so it is feasible toemployA2 to calculate the ultrasonic phase velocity spectrum.

As described before, the ultrasonic spectra can beobtained by using different methods to analyze the reflec-tion and transmission signals. Figure 6 demonstrates theultrasonic spectra of silicon-water suspensions with differentvolume fractions; it can be seen from (a), (b), and (c) that thetrends of different attenuation spectra are almost consistentand the attenuation grows gradually with the increasing con-centration, but the quantities of the through-transmissionmethod are larger than the other two methods (takingsilicon-water suspension with 12% volume fraction as anexample, the through-transmission method has an averagedeviation of 5.2% from the pulse-echo method and 7.1%from the reference reflection method). The larger errorof through-transmission can be attributed to the workingmodel, which employs two transducers. Although trans-ducer A and transducer B are of the same type, the consis-tency difference and even a slight deviation in installationcoaxality could affect the ultrasonic spectra measurements.The phase velocity spectra are displayed in Figure 6(d),which illustrates that the velocity in the suspension ishigher than the sound velocity (around 1497m/s) in waterat the same temperature (25°C), and it also shows anacoustic frequency dispersion that the velocities vary withultrasonic frequency but without exception grow with theincrease of concentration.

4. Inversion

For the sake of characterizing particles in suspensions byultrasonic spectra, first and foremost, the relationshipbetween particle concentration, particle size, and the ultra-sonic spectra should be established mathematically. Thedetermination of particle size distribution is not only relatedto the frequency dependence of experimental ultrasonic spec-tra but also the theoretical model and the inversion algorithm.

4.1. Theoretical Model. As soon as the ultrasonic attenuationand phase velocity spectra of the suspension have beenmeasured, it is imperative to convert them into a particlesize distribution by virtue of an appropriate theory. It isgenerally based on a mathematical treatment of the physi-cal process that occurs when an ultrasonic wave propa-gates through the medium. In solid-liquid two-phaseflow, the ECAH model is commonly used to model acous-tic propagation, which considers the scattering, absorption,and heat effects [27]. The following expression of the com-plex ultrasonic wave number can be used to interpret therelationship between characteristics of particle and theultrasonic spectra:

kkc

2= 1 + 3φ

jk3cR3 〠

∞

n=02n + 1 An, 10

Figure 2: The microscope image of silicon.

0 5 10 15 20 25 30 350

5

10

15

20

Image

D (μm)

Vol

ume f

ract

ion

(%)

Figure 3: The particle size distribution of silicon.

5Journal of Sensors

with

k = ω

cs ω+ jαs ω , 11

where αs and cs refer to the sample attenuation coefficient andphase velocity, respectively, which are associated with theultrasonic frequency in the two-phase measurement system;kc = 2π/λ is the compress wave number; j = −1 ⋅ φ refers to

the volume fraction of particles;R is the radius of the particles;n denotes the scattered level; andAn is known as the scatteringcoefficient, reflecting the interaction of ultrasonic waves andsuspended particles in media; it can be solved by combiningthe compression wave function and the boundary conditions.

It should be noted that particles are always in a polydis-perse condition, so the corresponding attenuation coefficientand velocity should be evaluated from the superposition ofcontribution from every single bin of particles and can be

5. Transmitter/receiver(transducer A)

4. Receiver (transducer B)

3. Measurement cell

6. Thermostat bath

7. Panametrics 5800PR

2. Panametrics 5077PR

1. Acquisition card (in host)

3

45

6

72

81

8. Signal display

(a)

Pulse transmitting &receiving device

Transducer A Transducer BSample

Pulse-receiving interface(amplifier)

Dual-channel digitizer

ExcitationA1, A2, A3A⁎

A⁎

10 mm10 mm 6.5 mm

A1A0A2A3

(b)

Figure 4: (a) Photo of the measurement setup configuration. (b) Sketch of the ultrasonic spectrometer used to measure the ultrasonicproperties of silicon aqueous suspensions.

6 Journal of Sensors

200 205 210 215 220 225 230

−2

−1

0

1

2

3

4

A3

A2

A1

A0

Reflection

t (�휇s)

Am

plitu

de (V

)

(a)

200 205 210 215 220 225 230

−2

−1

0

1

2

3

4

A⁎

Am

plitu

de (V

)

Transmission

t (�휇s)

(b)

Figure 5: Time-domain signals of ultrasound for samples: (a) reflection signals and (b) transmission signals.

2 3 4 5 6 7 8

20

40

60

80

100

f (MHz)

8%10%12%

Atte

nuat

ion

(Np·

m−1

)

(a)

2 3 4 5 6 7 8

20

40

60

80

100

f (MHz)

8%10%12%

Atte

nuat

ion

(Np·

m−1

)

(b)

2 3 4 5 6 7 8

20

40

60

80

100

f (MHz)

8%10%12%

Atte

nuat

ion

(Np·

m−1

)

(c)

2 3 4 5 6 7 81510

1515

1520

1525

1530

8%10%12%

f (MHz)

Phas

e vel

ocity

(m· s−

1 )

(d)

Figure 6: Ultrasonic spectra obtained by different measurement methods: (a) attenuation spectra with the pulse-echomethod, (b) attenuationspectra with the reference reflection method, (c) attenuation spectra with the through-transmission method, and (d) phase velocity spectrawith the reference reflection method.

7Journal of Sensors

expressed as follows:

αs =3φ2kc2

〠N

j=1

qjRj

3 〠∞

n=02n + 1 Re An Rj, ω , 12

1cs

−1cw

= 3φ2ωk2

〠N

j=1

qjR3j

〠∞

n=02n + 1 Im An Rj, ω , 13

where cw is phase velocity of water, which is usually known; ρis the equivalent density of suspensions; qj is the volume frac-

tion when the particle radius ranges from Rj to Rj+1; Re is theoperation of taking the real part of a complex array; and Im isfor the imaginary part operation.

The classical ECAH model is relatively comprehensiveand requires numerous material parameters as shown inTable 1 [28]. If there is no need to realize real-time analysis,the coefficient matrix can be calculated first to inverse theparticle size distribution.

4.2. Inversion Algorithm. The principle of PSD measurementis based on the properties of ultrasonic spectra generated bydifferent particle sizes and is determined by seeking thePSD which gives the best fit between theory and experimentaldata. The inversion algorithm for particle size can be dividedinto two types: the dependent model and independent model.For the former algorithm in inversion problems, it is usuallyassumed that the PSD follows a certain function form such asRosin-Rammler distribution (R-R), normal distribution, andlog-normal distribution. For example, the expression of R-Rdistribution function can be written as

dVdR

= k

R

R

R

k−1exp −

R

R

k

, 14

where dV/dR is the frequency distribution of particle vol-ume; R is the characteristic radius; and k is the distribution

parameter, namely, the narrowness index of the distribution,which is a dimensionless quantity.

The root mean square error can describe the differencebetween the simulated and the measured spectra as

ERMS =1N〠N

j=1

αs − αmαm

2, 15

where αm is the measured spectrum and αs is the simulatedone. The optimization process can be carried out until thecomputer terminates the search when the objective functionERMS reaches the minimum value, and then the optimal char-acteristic of R and k can be determined.

As a global optimization algorithm, the genetic algorithmis suitable for a dependent model, and it is a valid optimiza-tion approach based on the natural selection and the theoryof genetics, searching global optimization solution withoutany initialization information [23]. Thus, the authors havedeveloped an inversion algorithm based on the genetic algo-rithm before, where some crucial parameters, e.g., maximumgenerations, population size, and genetic operators (cross-over fraction and mutation fraction), were optimized for par-ticle size characterization with its improved form (see [25]).This method has already demonstrated some advantagescompared with other local optimization methods because ofglobal convergence, great robustness, suitable for parallelcomputing, and so on.

At the same time, an optimum regularization tech-nique has been proposed, as an independent model algo-rithm, which is improved by optimizing theregularization factors combined with mathematic opera-tions such as Generalized Cross Validation [29], to solvethe ill-conditioned equations and yield a higher stabilityof solution. For this purpose, the major equations (12)and (13) must be modified by altering calculation of coef-ficient matrix and the form of input ultrasonic spectra inorder to obtain more accurate particle size distribution;hence, the two equations are converted into linear equa-tions AF = G after discretization as follows:

where A is the coefficient matrix, F is the discrete vol-ume distribution of particle size to be determined, andG is the vector consisting of experimental ultrasonicattenuation or the phase velocity spectrum at different

frequencies, while the subscripts i and j indicate thenumber of discrete classifications of the radius for parti-cles and frequency for ultrasonic attenuation or phasevelocity, respectively.

3φ2 ⋅

ΔR

1k2cRj

3 〠∞n=0 2n + 1 Re An Rj, ωi ⋅ dR

Ai, j

⋅〠V ΔRi

Fi

= αs ωi

Gi

,

3φ2 ⋅

ΔR

1k2cωiRj

3 〠∞

n=02n + 1 Im An Rj, ωi ⋅ dR

Ai, j

⋅〠V ΔRj

F j

= 1cs ωi

−1cw

Gi

,16

8 Journal of Sensors

5. Results and Discussions

The ultrasonic attenuation/velocity spectra at the volumefraction of 12% silicon-water suspension were adopted to cal-culate the particle size distribution of samples. During thecalculations, the spectra measured by different methods wereanalyzed with different inversion algorithms. Figure 7 illus-trates the PSDs of the sample obtained from different ultra-sonic spectra; (a), (b), and (c) are inversed by attenuationspectra with different measurement methods, which show a

good agreement in the trend and distribution width withthe same inversion algorithm. In addition, the correspondingvolume median diameters (Dv50) and distribution width ofthe sample are shown in Table 2; it can be seen that theresults obtained by the pulse-echo method are much closerto the reference reflection method than the through-transmission method, and their volume median diametershave a relative deviation less than 2%, while the through-transmission method deviates from the other two methods,yielding an error still less than 15%. As described in the

Table 1: Physical parameters of water and silicon.

ρ (kg·m-3) c (m·s-1) α (Np·m-1) Cp (J·kg-1·K) τ (W·m-1·K) β (K-1) η (N·m-2)

Water 997 1483 2 53 × 10−14 f 2 4178.5 0.595 2 57 × 10−4 —

Silicon 2185 5968 2 6 × 10−22 f 2 729 1.6 1 35 × 10−6 3 09 × 1010

0 5 10 15 20 25 300

5

10

15

20

25

30

ORTGAImage

Vol

ume f

ract

ion

(%)

D (μm)

(a)

0 5 10 15 20 25 300

5

10

15

20

25

30

ORTGAImage

D (�휇m)V

olum

e fra

ctio

n (%

)

(b)

0 5 10 15 20 25 300

5

10

15

20

25

30

ORTGAImage

D (�휇m)

Vol

ume f

ract

ion

(%)

(c)

0 5 10 15 20 25 300

5

10

15

20

25

30

ORTGAImage

D (�휇m)

Vol

ume f

ract

ion

(%)

(d)

Figure 7: Particle size distribution with different ultrasonic spectra: (a) attenuation spectra with the pulse-echo method, (b) attenuationspectra with the reference reflection method, (c) attenuation spectra with the through-transmission method, and (d) phase velocity spectrawith the reference reflection method.

9Journal of Sensors

context, the amplitude of ultrasonic attenuation spectra withthe through-transmission makes a slight difference fromthose of other two methods, mainly causing the deviationof volume median diameters.

Figure 7(d) illustrates the particle size distribution gener-ated by the ultrasonic phase velocity spectra that also wellreflect the particle size information, and it can obviouslyreflect that the width of size distribution is narrower com-pared with those of ultrasonic attenuation spectra, while thevolume median diameter is comparative. It reveals that thewidths of particle size distributions obtained by the phasevelocity spectra method are relatively close to the imagemethod compared with the attenuation spectra methodwhenever the inversion algorithms are chosen.

Regarding the inversion algorithm, both GA and ORTalgorithms can recover relatively reasonable particle size dis-tribution, including the successful findings of the medianvalue and width of the particle system. As shown inTable 2, the volume median diameters inversed by GA andORT with the same measurement methods have a deviationfrom image analysis less than 10%. For the inversion of theattenuation spectrum, the ORT algorithm gives a wider par-ticle size distribution compared with GA algorithm, and thelatter is close to the result of the image method. But in thecase of phase velocity spectra, it shows that the width ofparticle size distribution obtained by the ORT algorithmis nearly the same as that of the image method and onlyslightly wider than that of the GA algorithm. It depictsan inconsistent adaptability to different ultrasonic spectrafor the ORT algorithm.

If the influence of spectral information is merely consid-ered, it can be found that there is an obvious relative variationfor the attenuation spectrum itself (see Figure 6), which isusually advantageous for analyzing the particle size distribu-tion. However, through further analysis of the coefficientmatrix (sensitive matrix), it can be seen that not only thespectral information but also the involved mathematicalproblems affect results. For instance, the condition numbersof the matrix after the singular value decomposition for theattenuation spectrum solution are particularly larger (up to3 × 109), about ten times of those for the phase velocity.Hence, it is likely to encounter a more serious ill-conditioned problem. This can help to explain that theinversed PSD of the attenuation spectrum is often too widewith ORT. However, the performance of the GA algorithmseems to be more stable on the whole; no matter what atten-uation spectra or phase velocity spectra are chosen, there is

no great deviation from the image analysis. As the GA algo-rithm here belongs to the dependent model, the calculationof particle size distribution must obey the Rosin-Rammlerdistribution function, which in turn exerts a strong con-straint for inversion. While for the ORT algorithm (indepen-dent model), there is no need for such a constraint ofdistribution function.

Back to the measurement method itself, the frequency-dependent ultrasonic spectra of the sample are measured ina basic similar fashion as mentioned in these three methods,and the analysis procedures are automated and straightfor-ward, while the differences and considerations in transducerinstallation, methodology principle, and signal analysis needto be stressed again. Firstly, the pulse-echo method and refer-ence reflection method use a single transducer to both trans-mit and receive the ultrasonic waves while the through-transmission method employs a pair of transducers to work.Given that case, it is obvious that the measurement device ofthe two former methods is relatively simple and commer-cially available regardless of caring for the transducer instal-lation coaxality. Secondly, the reference reflection methodand the through-transmission method belong to relativemeasurement; however, the pulse-echo method is an absolutemeasurement, which means that the pulse-echo method iseasier to operate without considering the calibration of refer-ence material. Finally, the pulse-echo method and referencereflection method must adopt the reflection signals duringspectral calculation, so it is important to select the properbuffer rod and bring a reliable impedance difference for accu-rate reflection signal acquisition, which need to be consideredcarefully in the practical applications.

6. Conclusions

The experimental setup was specially designed to provide anoninvasive way in synchronously acquiring ultrasonicreflection and transmission signals, which can be utilized tomeasure the frequency-dependent ultrasonic properties ofsuspensions. To obtain the particle size distribution, thepulse-echo method, reference reflection method, andthrough-transmission method were employed to extract theultrasonic attenuation and phase velocity spectra combinedwith the inversion algorithms GA and ORT. The particle sizedistributions inversed by these two inversion algorithms bothwell reflect the particle information and bring a relative devi-ation less than 10% about the volume median diameter with

Table 2: Comparison of the sample volume median diameter measured by ultrasonic spectra and image analysis (unit: μm).

DeviationMethod

Pulse-echo Reference reflection Through-transmissionApproach Dv50 Dv90 −Dv10 /Dv50 Dv50 Dv90 −Dv10 /Dv50 Dv50 Dv90 −Dv10 /Dv50

AttenuationORT 11.54(+1.5%) 0.98 11.43(+0.5%) 1.04 10.56(-7.1%) 1.03

GA 12.36(+8.7%) 0.41 12.21(+7.3%) 0.40 10.58(-6.9%) 0.52

Phase velocityORT / / 10.27(-9.7%) 0.42 / /

GA / / 11.62(+2.2%) 0.29 / /

Image N/A 11.37 0.32 11.37 0.32 11.37 0.32

10 Journal of Sensors

image analysis, which indicates that different methods have ahigh stability and reliability.

In brief, the reference reflection method is easy to obtainthe attenuation and velocity spectra at the same time. How-ever, as far as the results discussed in the present paper areconcerned, the procedure of inversion needs to pay moreattention to the adaptability of different algorithms to variousspectra, which often influences the width of particle size dis-tribution rather than the median diameter. Further, consid-ering the online measurement problem, the pulse-echomethod is very advantageous for the design of the conciseprobe since only one transducer is used, and the referencematerial can be avoided during the absolute measurement.Similarly, the through-transmission method is ideal formounting a pair of transducers on the tube wall to implementa real nondestructive testing, which is necessary in many pro-cess measurements.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

The present work is supported by the National NaturalScience Foundation of China (NSFC 51776129 and51676130), which is gratefully acknowledged.

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