Estimates of particle size distributions (PSDs) in solid-in-liquid suspensions can be obtained from measurements of ultrasonic wave attenuation. The technique is based on adaptively fitting theoretical wave propagation models to the measured data across a frequency range. These models break down at high solid concentrations and it is believed that this failure is due to the effective viscosity of the mixture in the vicinity of the particles being different from that of the continuous phase. This paper discusses PSD estimation when a number of different viscosity formulations are incorporated into the wave propagation model. The viscosity model due to Happel provides the best estimate of PSD in suspensions of medium concentration.
References
1.
R. E.
Challis
, M. J. W.
Povey
, M. L.
Mather
, and A. K.
Holmes
, “Ultrasound techniques for characterizing colloidal dispersions
,” Rep. Prog. Phys.
68
(7
), 1541
–1637
(2005
).2.
BS ISO 20998-1, “
Measurement and characterization of particles by acoustic methods—Part 1: Concepts and procedures in ultrasonic attenuation spectroscopy
,” British Standards Institute, p. 30
(2012
).3.
A. K.
Holmes
, R. E.
Challis
, and D. J.
Wedlock
, “A wide bandwidth study of ultrasound velocity and attenuation in suspensions: comparison of theory with experimental measurements
,” J. Colloid Interface Sci.
156
(2
), 261
–268
(1993
).4.
M. C.
Davis
, “Coal slurry diagnostics by ultrasound transmission
,” J. Acoust. Soc. Am.
64
(2
), 406
–410
(1978
).5.
D. J.
McClements
and M. J. W.
Povey
, “Scattering of ultrasound by emulsions
,” J. Phys. D: Appl. Phys.
22
(1
), 38
–47
(1989
).6.
A. K.
Holmes
, R. E.
Challis
, and D. J.
Wedlock
, “A wide-bandwidth ultrasonic study of suspensions: the variation of velocity and attenuation with particle size
,” J. Colloid Interface Sci.
168
(2
), 339
–348
(1994
).7.
A. K.
Hipp
, G.
Storti
, and M.
Morbidelli
, “Particle sizing in colloidal dispersions by ultrasound. Model calibration and sensitivity analysis
,” J. Phys. D: Appl. Phys.
15
(7
), 2338
–2345
(1999
).8.
S.
Meyer
, S.
Berrut
, T. I. J.
Goodenough
, V. S.
Rajendram
, V. J.
Pinfield
, and M. J. W.
Povey
, “A comparative study of ultrasound and laser light diffraction techniques for particle size determination in dairy beverages
,” Meas. Sci. Technol.
17
(2
), 289
–297
(2006
).9.
D. W.
Marquardt
, “An algorithm for least-squares estimation of nonlinear parameters
,” J. Soc. Ind. Appl. Math.
11
(2
), 431
–441
(1963
).10.
D. W.
Marquardt
, “Solution of nonlinear chemical engineering models
,” Chem. Eng. Prog.
55
(6
), 65
–70
(1959
).11.
R. E.
Challis
, A. K.
Holmes
, and V.
Pinfield
, “Ultrasonic bulk wave propagation in concentrated heterogeneous slurries
,” in Ultrasonic Wave Propagation in Non Homogeneous Media
, Springer Proceeding in Physics, edited by A.
Léger
and M.
Deschamps
(Springer
, Berlin
, 2009
), Vol. 128
, pp. 87
–98
.12.
M. L.
Mather
, R. E.
Challis
, A. K.
Holmes
, and A.
Kalashnikov
, “On the problem of the NDE of concentrated slurries
,” Rev. Prog. Quant. Nondestr. Eval., AIP Conf. Proc.
760
(1
), 1266
–1272
(2005
).13.
O.
Umnova
, K.
Attenborough
, and K. M.
Li
, “Cell model calculations of dynamic drag parameters in packings of spheres
,” J. Acoust. Soc. Am.
107
(6
), 3113
–3119
(2000
).14.
M. A.
Biot
, “Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range
,” J. Acoust. Soc. Am.
28
(2
), 168
–178
(1956a
).15.
M. A.
Biot
, “Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range
,” J. Acoust. Soc. Am.
28
(2
), 179
–191
(1956b
).16.
M. A.
Biot
, “Generalized theory of acoustic propagation in porous dissipative media
,” J. Acoust. Soc. Am.
34
(9A
), 1254
–1264
(1962a
).17.
M. A.
Biot
, “Mechanics of deformation and acoustic propagation in porous media
,” J. Appl. Phys.
33
(4
), 1482
–1498
(1962b
).18.
P. C.
Carman
, Flow of Gases Through Porous Media
(Butterworths
, London
, 1956
), p. 182
.19.
J. M.
Hovem
, “Viscous attenuation of sound in suspensions of high-porosity marine sediments
,” J. Acoust. Soc. Am.
67
(5
), 1559
–1563
(1980a
).20.
J. M.
Hovem
, “Erratum: Viscous attenuation of sound in suspensions of high-porosity marine sediments
,” J. Acoust. Soc. Am.
68
(5
), 1531
(1980b
).21.
A.
Einstein
, “Eine neue Bestimmung der Molekul-dimensionen (A new determination of molecular dimensions)
,” Ann. Phys.
324
(2
), 289
–306
(1906
).22.
V.
Vand
, “Viscosity of solutions and suspensions. I. Theory
,” J. Phys. Colloid Chem.
52
(2
), 277
–299
(1948a
).23.
V.
Vand
, “Viscosity of solutions and suspensions. II. Experimental determination of the viscosity-concentration function of spherical suspensions
,” J. Phys. Colloid Chem.
52
(2
), 300
–314
(1948b
).24.
V.
Vand
, “Viscosity of solutions and suspensions. III. Theoretical interpretation of viscosity of sucrose solutions
,” J. Phys. Colloid Chem.
52
(2
), 314
–321
(1948c
).25.
H.
Hasimoto
, “On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres
,” J. Fluid Mech.
5
(2
), 317
–328
(1959
).26.
A. A.
Zick
and G. M.
Homsy
, “Stokes flow through periodic arrays of spheres
,” J. Fluid Mech.
115
(1
), 13
–26
(1982
).27.
R. L.
Gibson
and M. N.
Toksoz
, “Viscous attenuation of acoustic waves in suspensions
,” J. Acoust. Soc. Am.
85
(5
), 1925
–1934
(1989
).28.
T. A.
Strout
, “Attenuation of sound in high-concentration suspensions: Development and application of an oscillatory cell model
,” Ph.D. thesis, Department of Chemical Engineering, University of Maine
, 1991
.29.
S.
Kuwabara
, “The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers
,” J. Phys. Soc. Jpn.
14
(4
), 527
–532
(1959
).30.
J.
Happel
, “Viscosity of suspensions of uniform spheres
,” J. Appl. Phys.
28
(11
), 1288
–1292
(1957
).31.
J.
Happel
, “Viscous flow in multiparticle systems: Slow motion of fluids relative to beds of spherical particles
,” AIChE J.
4
(2
), 197
–201
(1958
).32.
J.
Happel
and H.
Brenner
, Low Reynolds Number Hydrodynamics—With Special Applications to Particulate Media
(Prentice Hall
, Englewood Cliffs, NJ
, 1965
), p. 553
.33.
A. H.
Harker
and J. A. G.
Temple
, “Velocity and attenuation of ultrasound in suspensions of particles in fluids
,” J. Phys. D: Appl. Phys.
21
(11
), 1576
–1588
(1988
).34.
J. S.
Tebbutt
, “Ultrasonic absorption and phase velocity spectra colloids: Theory, simulation and measurement
,” Ph.D. thesis, Department of Physics, University of Keele
, 1996
.35.
P. S.
Epstein
and R. R.
Carhart
, “The absorption of sound in suspensions and emulsions. I. Water fog in air
,” J. Acoust. Soc. Am.
25
(3
), 553
–565
(1953
).36.
J. R.
Allegra
and S. A.
Hawley
, “Attenuation of sound in suspensions and emulsions: Theory and experiments
,” J. Acoust. Soc. Am.
51
(5B
), 1545
–1564
(1972
).37.
P.
Lloyd
and M. V.
Berry
, “Wave propagation through an assembly of spheres. Part I. V. Relations between different scattering theories
,” Proc. Phys. Soc.
91
(3
), 678
–688
(1967
).38.
R. E.
Challis
, J. S.
Tebbutt
, and A. K.
Holmes
, “Equivalence between three scattering formulations for ultrasonic wave propagation in particulate mixtures
,” J. Phys., D: Appl. Phys.
31
(24
), 3481
–3497
(1998
).39.
D. R.
Lide
, CRC Handbook of Chemistry and Physics: A Ready-Reference Book of Chemical and Physical Data
, 91st ed. (CRC Press
, London
, 2010
), p. 2610
.40.
G. W. C.
Kaye
and T. H.
Laby
, Tables of Physical and Chemical Constants
, 16th ed. (Longmans
, Harlow
, 1995
), p. 144
.41.
V. A. Del
Grosso
and C. W.
Mader
, “Speed of sound in pure water
,” J. Acoust. Soc. Am.
52
(5B
), 1442
–1446
(1972
).42.
M. C.
Smith
and R. T.
Beyer
, “Ultrasonic absorption in water in the temperature range 0°–80 °C
,” J. Acoust. Soc. Am.
20
(5
), 608
–610
(1948
).43.
R.
Al-Lashi
, “Novel approaches to ultrasonic particle sizing in suspensions with uncertain properties, and to the design of ultrasonic spectrometers
,” Ph.D. thesis, Electrical and Electronic Engineering Department, University of Nottingham
, 2011
.44.
F.
Luppé
, J.
Conoir
, and A.
Norris
, “Effective wave numbers for thermo-viscoelastic media containing random configurations of spherical scatterers
,” J. Acoust. Soc. Am.
131
(2
), 1113
–1120
(2012
).45.
A. K.
Hipp
, G.
Storti
, and M.
Morbidelli
, “Acoustic characterization of concentrated suspensions and emulsions. 1. Model analysis
,” Langmuir
18
(2
), 391
–404
(2002
).46.
A. K.
Hipp
, G.
Storti
, and M.
Morbidelli
, “Acoustic characterization of concentrated suspensions and emulsions. 2. Experimental validation
,” Langmuir
18
(2
), 405
–412
(2002
).47.
L. W.
Anson
and R. C.
Chivers
, “Ultrasonic scattering from spherical shells including viscous and thermal effects
,” J. Acoust. Soc. Am.
93
, 1687
–1699
(1993
).© 2014 Acoustical Society of America.
2014
Acoustical Society of America
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