浮力秤量法による粒径分布測定法の研究開発
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言語: English
出版者:
公開日: 2012-07-11
キーワード (Ja):
キーワード (En): Particle size distribution, buoyancy,
Allen region, stokes diameter, mean particle diameter
作成者: ロンダン, タムブン
メールアドレス:所属:

URL

https://doi.org/10.15118/00005092

Research and Development of the Particle Size Distribution Analyzer
by the Buoyancy Weighing-Bar Method

Division of Chemical and Materials Engineering
Graduate School of Engineering
Muroran Institute of Technology

Rondang Tambun

Research and Development of the Particle Size Distribution Analyzer
by the Buoyancy Weighing-Bar Method

byRondang Tambun

A Dissertation
Submitted in Partial Fulfillment of the Requirements for DegreeofDoctor of EngineeringinChemical Engineering

Division of Chemical and Materials Engineering
Graduate School of Engineering
Muroran Institute of Technology
Muroran, Japan
February 2012

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Abstract
The particle size distribution and mean particle size are the most importantcharacteristics in particulate or powder technology. Various approaches have been used tomeasure particle size distributions. For solidliquid systems, the particle size distributions
have been measured by the Stokes diameter. The Andreasen pipette method, sedimentation
balance method, centrifugal sedimentation method, etc. have been used to measure theparticle size distribution in suspensions. These methods measure the migration velocities ofparticles in solution, and then particle size is calculated using Stokes formula. However, allthese methods are time consuming and require special skills. On the other hand, a different
principle can be used to analyze the particle size distribution through microscopy, laser
diffraction/scattering method, and a Coulter counter method. These methods require numerous
samples to accurately determine the particle size distribution. Although the laser
diffraction/scattering and Coulter counter methods produce highly accurate results within ashorter time, they require extremely expensive equipment. Hence, a simple and cost effective
method to determine the particle size distribution is highly demanded.
The investigation of particle size distributions by the buoyancy weighing-bar method
which is a novel method in particle size distribution measurement have been done for several
samples particles of Japan Industrial Standard (Test Powders 1 and Test Powders 2),
Sumitomo 3M (glass beads K1, K37, S60HS), polystyrene beads (spherical), nylon beads(cylindrical), etc. The influence of solid concentration, influence of shape and size of
weighing bar and influence of vessel size also had been investigated in particle size
distribution measurement by the buoyancy weighing-bar method. The result data obtained by
the buoyancy weighing-bar method were comparable with the others particle size analyzer,
such as sedimentation balance method, Andreasen pipette method, Coulter counter method,
microscopy and the laser diffraction/scattering method. The buoyancy weighing-bar method issuitable for measuring the settling particle size distribution and the floating particle size
distribution. The buoyancy weighing-bar method could be used to measure the particle size
distribution in Stokes region and in Allen region. The particle size distribution obtained by thebuoyancy weighing-bar method could be used to estimate the smaller particle size by using
the Rosin-Rammler distribution. Theory was rediscussed and the new mean particle size wasdefined. Beside, the buoyancy weighing-bar method could be used to determine the mean
particle sizes for a ternary mixture and multi-component particles in a short time. This ability
is very useful in the powder industry.
Similar to other methods to analyze particle size, the buoyancy weighing-bar method is
accurate in an initially homogeneous suspension. Because the buoyancy weighing-bar method
apparatus can be made by hand, it is also economical. Additionally, the buoyancy
weighing-bar method is easier to be implemented compared to other sedimentation particle
analysis methods.
iContents
Abstract……………………………………………………………………………...
Contents ……………………………………………………………………………..
List of Figures ……………………………………………………………………….
List of Tables ………………………………………………………………………..

Page
i
ii
v
vii

List of Papers ………………………………………………………………………..
Chapter 1 Introduction……………………………………………………………
1.1 Background ……………………………………………………………………...
1.2. Principle of BWM ………………………………………....................................
1.2.1 Settling particle……………………………………………………………
1.2.2 Floating particle…………………………………………………………...

viii
1
1
2
2
4

1.3 Calculation rules of PSDs by the BWM ………………………………………...
1.3.1 Size interval………………………………………………….....................
1.3.2 Time interval………………………………………………………………

6
6
6

1.3.3 Apparent mass interval……………………………………………………
1.3.4 Size ratio ……………………………………………………......................
1.4 Previous research …………………………………………………......................
1.5 Objective of the experiment………………………………………......................

6
7
7
7

Chapter 2 Graphical analogy of particle size distribution among Andreasen
pipette, sedimentation balance, fluidization curve and buoyancy
weighing-bar methods……………………………………………….
2.1 Introduction ……………………………………………………………………...
2.2 Theory …………………………………………………………………………...
2.2.1 Graphical analogy of PSD between the Andreasen pipette and fluidization

14
14
14

curve methods ………………………………….........................................
2.2.1.1 Andreasen pipette method ………………………………………...
2.2.1.2 Fluidization curve method …………………………......................
2.2.2 Graphical analogy of PSD between the BWM and the sedimentation balance

14
14
15

method ……………………………………………………………………
2.2.2.1 BWM ……………………………………………………………..

15
16

2.2.2.2 Sedimentation balance method …………………………………...
2.3 Material and Method …………………………………………………………….
2.4 Results and Discussion ………………………………………………………….
2.5 Conclusions ……………………………………………………………………...

16
16
17
17

Chapter 3 Measurement and estimation of settling particle size distribution by
the buoyancy weighing-bar method and the Rosin–Rammler
distribution ................................................................................................
3.1 Introduction ……………………………………………………………………...
3.2 Theory …………………………………………………………………………...

26
26
26
ii

3.3 Material and Method …………………………………………………………….
3.4. Results and discussion ………………………………………………………….
3.4.1 Combination validity of Rosin-Rammler distribution …………………...
3.4.2 Estimation accuracy ……………………………………………………...
3.4.3. Suspended solid in spring water ……………………………...................
3.5 Conclusions ……………………………………………………………………...

27
28
29
30
30
31

Chapter 4 Graphical and numerical determinations of mean particle size ofsettling particles by the buoyancy weighing-bar method …………..
4.1 Introduction ……………………………………………………………………...
4.2 Theory …………………………………………………………………………...
4.2.1 Schematic diagram of particle sedimentation ……………………………..
4.2.2 Definition of maximum particle size and velocity mean particle size of all

42
42
42
42

the particles ……………………………………………………………….
4.2.3 Velocity mean particle sizes of all the particles and from a given particle size
to a minimum size………………………………………………………...

44

4.2.4 Velocity mean particle size in the BWM………………………………….
4.2.5 Particle size distributions and velocity mean particle sizes in multicomponent mixtures………………………………………………………
4.3 Material and Method …………………………………………………………….

46

4.3.1 Sample particles …………………………………………………………...
4.3.2 Experimental apparatus …………………………………………………...
4.3.3. Experimental conditions and methods ……………………………………
4.4 Results and discussion …………………………………………………………..
4.5 Conclusions ……………………………………………………………………...

49
49
49
50
51

Chapter 5 Measurement of the floating particle size distribution by the buoyancy
weighing-bar method ………………………………………………….
5.1 Introduction ……………………………………………………………………...
5.2 Theory …………………………………………………………………………...
5.3 Materials and Methods …………………………………………….....................

61
61
61
62

5.3.1 PSD measurement of glass beads GBL 100 (JIS Test Powders 2) and hollow
glass beads K1 and S60HS (Sumitomo 3M) by the BWM …….................

62

5.3.2 Influence of the weighing bar and vessel in the BWM on size distribution
measurement of floating particle………………........................................
5.4 Results and Discussion ………………………………………………………….
5.4.1 PSD measurement of glass beads GBL 100 (JIS Test Powders 2) and hollow
glass beads K1 and S60HS (Sumitomo 3M) by the BWM …………........
5.4.1.1 PSD of glass beads GBL 100, JIS Test Powders 2 in heavy liquid ….
5.4.1.2 PSD of hollow glass beads K1, Sumitomo 3M …………..………
5.4.1.3 PSD of hollow glass beads S60HS, Sumitomo 3M ……………......
5.4.2 Influence of the weighing bar and vessel in the BWM on size distribution

45

47
49

63
64
64
64
64
65
iii

measurement of floating particle……………….........................................
5.4.2.1 Accuracy and repeatability of PSD measurements ……………….
5.4.2.2 Influence of weighing bar length …………………………………
5.4.2.3 Influence of weighing bar diameter ………………………………
5.4.2.4 Influence of vessel size…………………………………………...
5.4.2.5 Influence of weighing bar shape………………………………….

65
65
66
66
66
67

5.4.2.6 Influence of vessel shape and weighing bar position .....................
5.4.2.7 Influence of the sectional area ratio ………………………………
5.5 Conclusions ……………………………………………………………………...

67
67
68

Chapter 6 Size distribution measurement of floating particles in the Allen region
by the buoyancy weighing-bar method ………………………………
6.1 Introduction ……………………………………………………………………...

79
79

6.2 Theory …………………………………………………………………………...
6.3 Materials and Methods …………………………………………….....................
6.4. Results and Discussion …………………………………………………………

79
80
80

6.4.1. Spherical particles ………………………………………….....................
6.4.2. Cylindrical particles ………………………………………......................
6.5 Conclusions……………………………………………………………………...
Chapter 7 Conclusions…………………………………………………..................

80
81
81
87

7.1 Settling particles ………………………………………………………………...
7.2 Floating particles ………………………………………………………………..
References…………………………………………………………………………
Acknowledgements ………………………………………………………………….

87
88
90
95

iv

List of Figures
Figure 1.1
Figure 1.2
Figure 1.3

Determination of PSD by plot of pressure drop P vs. superficial velocity u in
fluidization
Schematic diagram of particle settling
Determination of settling PSD using the BWM

Figure 1.4
Figure 1.5
Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4

Schematic diagram of particle floating
Determination of the floating PSD using the BWM
Graphical determination of PSD in Andreasen pipette method
Graphical determination of PSD in settling balance method
Schematic diagram of experiment apparatus
Weighing bar (Slit-cylinder type)Figure 2.5
Figure 2.6
Figure 2.7

Apparent mass of weighing bar as a function of time (glass beads J-400)
PSD of glass beads J-400
PSD of white fused alumina no.2, JIS Test Powders 2

Figure 2.8
Figure 2.9

PSD of white fused alumina no.3, JIS Test Powders 2
PSD of particle mixture of white fused alumina no.2 and no.3 (40%:60%), JIS
Test Powders no.2
Apparent mass of weighing bar as a function of time

Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6

PSDs of JIS Test Powders 2, No.4 (White fused alumina)
PSDs of non–spherical silica sand Class 3 (JIS Test Powders 1)
PSDs of the non–spherical calcium carbonate (heavy) Class 16, JIS Test
Powders 1
PSDs of the JIS Test Powders 1, class 4 (Talc)
PSDs of magnesite (China)

Figure 3.7
Figure 3.8
Figure 3.9
Figure 3.10

PSDs of soft–burned magnesia (China)
PSDs of spherical glass beads GBL 30 (JIS Test Powders 2)
PSDs of the JIS Test Powders 1, class 8 (KANTO (Japanese) loam)
PSDs of the JIS Test Powders 1, class 5 (Fly ash)

Figure 3.11
Figure 3.12

Rosin-Rammler plot (JIS Test Powders 1, class 3 (Silica sand))
PSDs of JIS Test Powders 1, class 3 (Silica sand)Figure 3.13

PSDs of JIS Test Powders 1, class 10 (Fly ash) and class 11 (KANTO(Japanese) loam)
PSDs of JIS Test Powders 1, class 17 (Calcium carbonate, heavy)
PSDs of suspended solid in spring water
Schematic diagram of the change in ideal particle settling over time.
Mass ratio vs. settling time in ideal particle settling

Figure 3.14
Figure 3.15
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4

Determination of the particle size distribution and mean particle size of aternary mixture using the buoyancy weighing bar method
Determination of the particle size distribution and mean particle size ofv

Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8

multi-component particles using the buoyancy weighing-bar method
Relationship between the mean particle size and mass frequency curve
Apparent mass of the weighing bar as a function of time with glass beads (GBL
30, JIS Test Powders 2)
Particle size distributions of glass beads (GBL 30, GBL 40, and GBL 60, JIS
Test Powders 2)

Figure 5.1
Figure 5.2

Apparent mass of the weighing bar as a function of time with the ternary
mixture (glass beads GBL 60, GBL 40 and GBL 30, JIS Test Powders 2)
Particle size distributions of a ternary mixture of glass beads (GBL 60, GBL 40
and GBL 30, JIS Test Powders 2)
Schematic diagram of the experimental apparatus, weighing bar and vessel
Schematic diagrams of the weighing bars

Figure 5.3
Figure 5.4
Figure 5.5

PSDs of glass beads GBL 100, JIS Test Powders 2 in heavy liquid
PSDs of hollow glass beads K1, Sumitomo 3M
PSDs of hollow glass beads S60HS, Sumitomo 3M

Figure 5.6
Figure 5.7
Figure 5.8

PSDs measured by microscopy, laser diffraction/scattering method and BWM
Time course change in the apparent mass of the weighing bar using the 1000 ml
graduated cylinder
Influence of rod length on PSD (vessel diameter = 65 mm)

Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12
Figure 5.13
Figure 5.14

Influence of rod diameter on PSD (vessel diameter = 65 mm)
Influence of rod diameter on PSD (vessel diameter = 38 mm)
Influence of vessel size on PSD
Influence of weighing bar shape on PSD
PSDs measured in the square vessel
Particle sizes as a function of sectional area ratio

Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4

Apparent mass of the weighing bar as a function of time (spherical particle)
Particle size distributions of spherical particle
Apparent mass of the weighing bar as a function of time (cylindrical particle)
Particle size distributions of cylindrical particle

Figure 4.9vi

List of Tables
Table 1.1
Table 2.1
Table 3.1
Table 4.1

History of several measurements methods in PSD
Particle size of sample particles
Properties of sample particles (granted by Japan Industrial Standard)
Particle size of the sample particles

Table 4.2

Mean particle size of a ternary mixture of glass beads GBL 60, GBL 40, andGBL 30Sizes of weighing bar and vessel
Particle size, standard deviation and variation coefficientParticle and liquid properties

Table 5.1
Table 5.2
Table 6.1

vii

List of Papers<Papers>(1) Ohira, Y., K. Furukawa, R. Tambun, M. Shimadzu and E. Obata; Buoyancy
Weighing-bar Method: - A Particle Size Distribution Measurement using New Settling
Method -, Journal of the Sedimentological Society of Japan, 69, 17-26 (2010.9)(2) Tambun, R., T. Motoi, M. Shimadzu, Y. Ohira and E. Obata; Size Distribution
Measurement of Floating Particles in the Allen Region by a Buoyancy Weighing-Bar
Method, Advanced Powder Technology, 22, 548-552 (2011.7)(3) Ohira, Y., K. Nakano, R. Tambun, M. Shimadzu, M. Ohta and E. Obata; Size
Distribution Measurement of Floating Spherical Particles by the Buoyancy
Weighing-Bar Method, Kagaku Kogaku Ronbunshu, 37, 310-316 (2011.7)(4) Tambun, R., M. Shimadzu, Y. Ohira and E. Obata; Definition of the New Mean Particle
Size based on the Settling Velocity in Liquid, Journal of Chemical Engineering of Japan,
in press(5) Tambun, R., K. Nakano, M. Shimadzu, Y. Ohira and E. Obata; Sizes Influences of
Weighing Bar and Vessel in the Buoyancy Weighing-Bar Method on Floating Particle
Size Distribution Measurements, Advanced Powder Technology, in press(6) Tambun, R., Y. Ohira, K. Furukawa, M. Hirayama, M. Shimadzu and E. Obata;Measurement and Estimation of Particle Size Distributions by the Buoyancy
Weighing-Bar Method and the Rosin-Rammler Distribution at Construction Sites,
Particuology, submitted<Presentations>(1) Tambun, R., Y. Ohira and E. Obata; Size Distribution Measurement of Settling Particles
by a Buoyancy Weighing-Bar Method, JSED 2010, pp.90-91, Muroran, JAPAN (2010.3)(2) Tambun, R., Y. Ohira and E. Obata; Graphical Analogy of Particle Size Distribution
among Andreasen Pipette, Settling Balance, Fluidization-Curve and Buoyancy
Weighing-Bar Method, 13th Asia Pacific Confederation of Chemical Engineering
Congress (APCChE 2010), CD-ROM, Chinese Taipei, TAIWAN-CHINA (2010.10)(3) Tambun, R., M. Shimadzu, Y. Ohira and E. Obata; Influence of Solid Concentration on
Particle Size Distribution Measurement by the Buoyancy Weighing-Bar Method, JSED2011, pp.86-87, Muroran, JAPAN (2011.3)･･･JSED Poster Prize(4) Nakano, K., R. Tambun, M. Shimadzu, Y. Ohira, M. Ohta and E. Obata; Application of
the Buoyancy Weighing-Bar Method to Floating Particle Size Measurement, JSED 2011,
pp.88-89, Muroran, JAPAN (2011.3)(5) Tambun, R., Y. Ohira and E. Obata; Graphical and Numerical Determinations of Mean
Particle Diameter by Buoyancy Weighing Bar Method, 8th European Congress ofChemical Engineering (ECCE 2011), CD-ROM, Berlin, GERMANY (2011.9)
viii

(6) Kato, H., R. Tambun, M. Shimadzu, Y. Ohira and E. Obata; Measurement of Particle
Density Distribution by the Buoyancy Weighing-bar Method, SCEJ 43th Autumn
Meeting, Nagoya, JAPAN (2011.9)ixChapter 1
Introduction
1.1 Background
Various approaches have been used to measure particle size distributions [17]. The
particle size distribution (PSD) is calculated based on several models, most often by number,by mass/volume and by surface. The modern sciences of PSDs have been published since thebeginning of the 20th century. The short history of several modern sciences of PSDs
measurements were shown in Table 1.1.
For solidliquid systems, PSDs have been measured by the Stokes diameter. The
Andreasen pipette method [16], sedimentation balance method [11], centrifugal sedimentation
method [12], etc. have been used to measure the PSD in suspensions. These methods measure
the migration velocities of particles in solution, and then particle size is calculated using
Stokes formula. However, all these methods are time consuming and require special skills. On
the other hand, a different principle can be used to analyze the PSD through microscopy [13],laser diffraction/scattering method [9], and Coulter counter method [19]. These methods
require numerous samples to accurately determine the PSD. Although the laser
diffraction/scattering and Coulter counter methods produce highly accurate results within ashorter time, they require extremely expensive equipment. Hence, a simple and cost effective
method to determine the PSD is highly demanded.
Obata et al. have reported five measurements for PSD by fluidization in the laminar toturbulent flow region [1-7,14], and one of them can be obtained by a graphical measurement
using a fluidization curve, i.e. the superficial velocity vs. pressure drop is plotted [4,5]. The
fluidization curve is constructed using the same algorithm as the sedimentation (Odén)
balance method [15].
The sedimentation method, which uses gravity, can simply and economically measure
the PSD, but it is time consuming. Bardet and Young have reported that the buoyancy method
can measure the PSD [10]. The buoyancy method determines the PSD by measuring thebuoyancy of a weighing ball. We tried the buoyancy method, but we were unable to measure
the PSD because the particles accumulated on the weighing ball. To prevent particle
accumulation, we used a weighing bar.
Obata et al. have strived to develop a new method to measure the PSD using thebuoyancy weighing bar method (BWM) [8]. In BWM, the change in suspension density due
to particle migration is measured by weighing buoyancy against a weighing bar hung in asuspension. Although the buoyancy method uses incremental analysis, the BWM uses
cumulative analysis. The principles of these two methods differ. In BWM the PSD iscalculated using the length of the weighing bar and the change in the apparent mass of theweighing bar with time.
1

Although it is difficult to construct a handmade Odén balance, assembling a handmade
weighing bar is easy. The proposed method is accurate just as the Andreasen pipette method
and economical. Thus, the BWM is suitable to be used in developing countries.
1.2 Principle of BWM
1.2.1 Settling particle
PSD can be classified as either a cumulative or frequency function [17]. The distribution
obtained in this method is the same cumulative function used in the Odén balance method[15] and manometric method. Although sedimentation curves are not really relevant to
fluidization curves, they are analogous to plots of mass vs. settling time and plots of pressure
drop vs. superficial velocity. Figure 1 plots the pressure drop ΔP throughout the entire bed
against the superficial velocity u [4-6]. If the superficial velocity u is u3, then:(    ) M 0 g D( x )
dP
(1.1)
P  [ P
][
]u
 0Y
P A
100
du
[

(  P   ) M 0 g D( x )
dP
 XY
][
]  0X , u
du
P A
100

where M0, A, and D(x) are the total particle mass, cross-sectional area of the bed, andcumulative mass percentage undersize of particle size x, respectively. Figure 1.1 graphically
depicts the PSD using the fluidization curve.
The volume of the submerged weighing bar is VB  Ah , where A is the cross-sectional
area of the weighing bar and h is the length of the submerged weighing bar in the suspension.
The densities of dispersion liquid containing the dispersant and particles are denoted as ρL,and ρP, respectively. The initial solid concentration of the suspension is C0[kg-solid/m3-suspension].
Figure 1.2 illustrates the schematic diagram of particle settling. Figure 1.2(a) showsthat initial buoyant mass of the submerged weighing bar depended on the particles between
the top of weighing bar and the bottom of that in the initial suspension.
At the settling time t  0 , the initial density of the suspension ρS0 is: S0   L 

C0

P P   L (1.2)

Because the initial buoyant mass of the submerged weighing bar WB0 depends on the particles
in the suspension from the suspension surface to depth h, WB0 can be defined as:
WB0  VB S0(1.3)

Then the apparent mass of the weighing bar in the initial suspension is:
GB0  VB  B  WB0  VB (  B  S0 )(1.4)

where ρB is the density of the weighting bar.
Figure 1.2(b) shows that concentration of suspension C decreases with time, because large
2particle have settled. The density of suspension ρSt , the buoyant mass of weighing
bar WBt and the apparent mass of the bar GBt in suspension at t = t is given by the following
equations:ρ St  ρ L  ρP  ρL 
ρP

C

(1.5)

WBt VB . ρSt(1.6)

GBt VB . ρB WBt VB . ρB VB . ρSt VB  ρB  ρSt (1.7)

Figure 1.2(c) shows that concentration of suspension C is finally zero, because small particles
also have settled. The final density of suspension S∞, the final buoyant mass of weighing bar
WB and the final apparent mass of the bar GB∞ in suspension at t = tmin is given by thefollowing equations:S   L(1.8)

WB VB . ρLGB VB . ρB WB VB  ρB  ρL 

(1.9)
(1.10)

Eq.(1.11) shows the mass balance of particles in suspension [17].

C0  C  C0 

xmax

xi

f ( x)dx C0 

xi

xmin

v( x)tf ( x)dxh(1.11)

From Eqs.(1.3), (1.6), (1.9) and (1.11),

W0  W  (W0  W ) 

xmax

xi

v( x)tf ( x)dx
xmin hf ( x)dx (W0  W ) xi(1.12)

where v(x) is the settling velocity of the particle, f(x) is the mass frequency of the particle size
x. Differentiate Eq.(1.12) with respect to the time t, we obtain:xi v( x)dW
 (W0  W ) f ( x)dx
xmin hdt(1.13)

From Eqs.(1.12) and (1.13),
 dWBt 
WBt  WRt  t dt 

(1.14)xmax
f ( x)dx.
xi

where WRt is the mass of particles greater than particle size x, W0  (W0 W ) Combining Eqs. (1.7) and (1.14) produces: dGBt  dG 
t  G Rt   Bt t dt 
 dt 

G Bt VB . ρ B WRt (1.15)dG Bt
dW  Bt , because, the decreasing mass of weighing bar isdt
dt
corresponding to the decreasing buoyant mass of the weighing bar. Value of GRt is calculated
where GRt VB . ρB WRt , andfrom tangent line based on Eq.(1.15). The cumulative mass oversize R(x) is:xmax
G GB 0
1 D( x)R( x)  
f ( x)dx  Rt
xi
GB  GB 0

(1.16)

Particle size x can be expressed by the following equation using Stokes formula:3

x

18 L v( x)g( P   L )(1.17)

where g is the gravitational acceleration and μL is the viscosity of the dispersion liquidcontaining the dispersant. The sedimentation velocity of the particles v(x) can be calculated by
Eq. (1.18):v( x) ht

(1.18)

where h is the submerged length of the weighing bar and t is the settling time. Particle size x
given by Eq. (1.17) is the Stokes diameter. Thus, the theory of BWM is the similar to that for
sedimentation balance analysis [15].
Figure 1.3 shows the calculation method of settling PSD. The upper right graph depicts
the change in the apparent mass of weighing bar GB with time, while the lower right graphshows the relationship between time and the reciprocal of the particle size. From Eqs. (1.17)
and (1.18), time is proportional to the second power of the reciprocal of particle size. Hence,in this method, particle size x can be calculated at time t while GRt can simultaneously becalculated from tangent line based on Eq. (1.15). Mass percentage undersize D(x) can becalculated by Eq. (1.16). Thus, the upper left graph illustrates the PSD from the calculatedparticle size x and D(x).
1.2.2 Floating particle
Theoretically, the BWM can be applied not only to settling particles, but also to floating
particles, including bubbles and liquid droplets [18]. However, the latter has not been verified.
Let us assume the particles are uniformly dispersed in suspension. Schematic diagram ofparticle floating is shown in Figure 1.4. As shown in Figure 1.4(a), the initial buoyant massof the submerged weighing bar WB0 depended on the particles between the top of theweighing bar and the bottom of that in the suspension. The initial density of suspension ρS0,
the initial buoyant mass of the weighing bar WB0 and the initial apparent mass of the bar GB0
in suspension at t  0 is given by the following equations:C S0   L  0  P   L (1.19)P

WB0  VB S0

GB0  VB  B  WB0  VB (  B  S0 )(1.20)(1.21)

where the liquid density is ρL, the particle density is ρP, the initial concentration of suspension
is C0 [kg–solid/m3–suspension], the density of the weighing bar in suspension is ρB, thevolume of the weighing bar is VB. As shown in Figure 1.4(b), concentration of suspension C
decreases with the time, because large particles have floated. The density of suspension ρS, thebuoyant mass of the weighing bar WBt and the apparent mass of the bar GBt in suspension att  t is given by the following equations:4S   L CP P   L (1.22)

WBt  VB S(1.23)

GBt  VB  B  WBt  VB (  B  S )(1.24)As shown in Figure 1.4(c), concentration of suspension C is finally zero, because small
particles also have floated. The final density of suspension ρS∞, the final buoyant mass of theweighing bar WB∞ and the final apparent mass of the weighing bar GB∞ in suspension att   is given by the following equations:S   L

(1.25)

WB  VB  L(1.26)

GB  VB  B  WB  VB (  B   L )(1.27)

Eq.(1.28) shows the mass balance of particles in suspension [17].

C0  C  C0 

xmax

xi

f ( x)dx C0 

xi

xmin

v( x)tf ( x)dxh(1.28)

From Eqs.(1.20), (1.23), (1.26) and (1.28),

W0  W  (W0  W ) 

xmax

xi

v( x)tf ( x)dx
xmin hf ( x)dx (W0  W ) xi(1.29)

where v(x) is the floating velocity of the particle, f(x) is the mass frequency of the particle size
x. Differentiate Eq.(1.29) with respect to the time t, we obtain:xi v( x)dW
 (W0  W ) f ( x)dx
xmin hdt

From Eqs.(1.29) and (1.30),
 dWBt 
WBt  WRt  t dt 

(1.30)(1.31)

The apparent mass of the submerged weighing bar GBt is given by Eq.(1.24). It is gradually
decreases from GB0 to GB∞. The volume and the density of the submerged weighing bar are
constant value. Differentiate Eq.(1.24) with respect to the time t, we obtain:

dGBt
dWBt

dt
dt

(1.32)

Therefore, from Eqs.(1.24), (1.31) and (1.32), we obtain: dG 
 dG 
GBt  VB  B  WRt   Bt t  GRt   Bt t(1.33)
 dt 
 dt 
where GRt  VB  B  WRt . Value of GRt calculates from tangent line based on Eq.(1.33). The
cumulative mass percentage oversize is:xmax
G GB 0
1 D( x)R( x)  
f ( x)dx  Rt
xi
GB  GB 0

(1.34)

Particle size x is given by using Stokes formula. The floating velocity of the particles
5

v(x) can be calculated by Eq. (1.34):v( x) ht

(1.34)

where h is the submerged length of the weighing bar and t is the floating time.
Figure 1.5 shows the calculation method of floating PSD. The lower right graph depicts thechange in the apparent mass of weighing bar GB with time, while the upper right graph shows
the relationship between time and the reciprocal of the particle size. This theory for thefloating particles measured by the BWM is the similar to that for settling particles.
1.3 Calculation rules of PSDs by the BWM
There are some basics calculation rules for research data obtained. The calculation rules
are necessary to obtain the more accuracy result. All these calculation rules give the close
results, but have the different difficulty. The basic calculation rules are size interval, timeinterval, apparent mass interval and size ratio. Following is the procedure of those
calculations.
1.3.1 Size interval
1.
2.
2.
3.
4.

Calculate the size.
Set a size interval.
Cut the data based on size interval.
Calculate the PSD.
Check the undersize fraction from minimum size. If undersize fraction is smaller thanprevious data, then its data is cut. If undersize fraction is over 1, then size fraction is 1.

5. Data in Allen region have to be cut. (Sphere-shaped: Re > 6, Others: Re > 2)
1.3.2 Time interval
1. Set a time interval.
2. Cut the data based on time interval.
3. Calculate the PSD.
4. Check the undersize fraction from minimum size. If undersize fraction is smaller thanprevious data, then its data have to be cut. If undersize fraction is over 1, then size fractionis 1.
5. Data in Allen region have to be cut. (Sphere-shaped: Re > 6, Others: Re > 2)
1.3.3 Apparent mass interval
1.

Set a apparent mass interval.
6

2.
3.
4.

5.

Cut the data based on apparent mass interval.
Calculate the PSD.
Check the undersize fraction from minimum size. If undersize fraction is smaller thanprevious data, then its data have to be cut. If undersize fraction is over 1, then size
fraction is 1.
Data in Allen region have to be cut. (Sphere-shaped: Re > 6, Others: Re > 2)

1.3.4 Size ratio
1.
2.
3.

Calculate the size.
Set a size ratio interval.
Cut the data based on size ratio.

4.
5.

Calculate the PSD.
Check the undersize fraction from minimum size. If undersize fraction is smaller thanprevious data, then its data is cut. If undersize fraction is over 1, then size fraction is 1.

6.

Data in Allen region have to be cut. (Sphere-shaped: Re > 6, Others: Re > 2)

1.4 Previous research
There are two the important experiments had been done by using the BWM, they are:
1. New measurement of PSD by the BWM [8]. In this experiment, Obata et al.
experimentally investigated the applicability of BWM to measure PSDs. They
investigated that the BWM was a easy, useful and economical method in PSD
measurement.
2. Measurement of the floating PSD by the BWM [18]. In this experiment, Motoi et al.
concluded that the BWM could measure the floating PSDs.
1.5 Objective of the experiment
The purpose of this study is to develop the BWM as a novel method in PSD
measurement. In this study, we investigated many factors affecting result of the PSD, such as:
1. Graphical analogy of PSD among Andreasen pipette method, settling balance method,
fluidization curve and the BWM.
2. Measurement and estimation of PSD by the BWM and the Rosin-Rammler distribution.
3. Graphical and numerical determinations of mean particle size of settling particles by the
BWM.
4. Influence of solid concentration on PSD measurement by the BWM.
5. Influence of the weighing bar and vessel in the BWM on floating PSD measurement.
6. Size distribution measurement of floating particles in the Allen region by the BWM.
7

Nomenclature18 L), m2･sh(  P   L ) gaconstant ( ACcross-sectional area of weighing bar or fluidization bed, m2
solid concentration of suspension, kg/m3

D(x)
f(x)

mass percentage undersize of particle size x, %mass frequency of the particle size x, m-1

g
GBt
GRt

gravitational acceleration, m/s2
mass of weighing bar at t = t, kg
VB B  WRt , kghsubmerged length of weighing bar, mxv ( x) ρ L
Reynolds number ( 
), –
μL

Re
R(x)
t
u

mass percentage oversize of particle size x, %time, sv(x)VBsuperficial velocity, m/s
settling velocity of particle size x, m/s
submerged volume of weighing bar, m3

WBbuoyant mass of the submerged weighing bar in the suspension, kg

WRt

WB0 

x
μL

particle size, mliquid viscosity, Pa･sρLliquid density, kg/m3

xmax

x

f ( x)dx , kg

density of weighing bar, kg/m3
ρBρP
particle density, kg/m3
ρS
density of suspension, kg/m3
Subscripts
max
maximumminminimum
0∞

initial t= 0
infinity t = ∞80Pressure drop, ΔP [Pa]Undersize fraction, D(x) [–]

ΔPo

1.0

xmax xi

x3

x2

x1

xmin

0

Particle size, x [m]YuXumin u1

u2

dΔP
 XY
du

u3

ui

umax

Superficial velocity, u [m/s]xminu  ax 2

xmax

Figure 1.1 Determination of PSD by plot of pressure drop P vs.
superficial velocity u in fluidization.

h=0

h=h

(a) t = 0(b) t = t(c) t = tmin

Figure 1.2 Schematic diagram of particle settling

9

Apparent mass, GB [kg]Oversize fraction, R(x) [–]

GB∞

1.0

0

xmax

x1

x2

Particle size, x [m]

GBt
GRt
GR2

GR1
GB0

xt xminT

tmax t1

t2

1
xmax
1
x1

tmin

t

Settling time, t [s]ti 

18 μ L hρ  ρ gx
p

L

2
i

1
a  
 xi 

2

1
x2
1
xt
1
xmin

Figure 1.3 Determination of settling PSD using the BWM.

h=h

h=0
(a) t = 0(b) t = t(c) t = tmin

Figure 1.4 Schematic diagram of particle floating

10

xn-1
xi-1
x3-1

ti 

x2-1

0

x1

Particle size, x [m]
x2
x3
xi

xn

x1>x2>･･･>xi>･･･>xn

Apparent mass, GB [kg]Oversize fraction, R(x) [–]

1

2

x1-1
GB0
GR1

 G  G0 
R   Ri
 G  G0 

1
18L h a (  L   P ) gxi 2
 xi 

t1

t2

t3

ti
Floating time, t [s]

tn

GR2
GR3
GRi

 dG 
GBt  GRt   Bt t dt GRn
GB∞

Figure 1.5 Determination of the floating PSD using the BWM.

11

Table 1.1 History of several measurements methods in PSD
MethodSedimentation
Sedimentation

Author, title and journal

Year

S. Odén; Eine neue methode zur mechanischen Bodenanalyse,
1915
Internationale Mitteilung fur Bodenkunde, 5, 575–311
S. Odén; Eine neue methode zur bestimmung der körnerverteilung
in suspensionen, Kolloid

Zeitschrift, 18, 33–47

1916

Manometric

G. Weigner; Ist es der chemischen grogtechnik zur zeit schon
msglich, ffir die, Landw. Versuch. Stat, 91, 41

PipetteG. W. Robinson; A new method for the mechanical analysis of soils
and other dispersions, Journal of Agricultural Science, 2, 1922
306–321Microscopy

G. Martin, C. E. Blyth and H. Tongue; Trans. Ceram. Soc, 23, 61

Sedimentation

S. Odén; The size distribution of particles in soils and theexperimental methods of obtaining them, Soil Science, 19 (1), 1924
1–35

Hydrometer

W. J. Kelley; Determination of distribution of particle size,
1924
Industrial and Engi. Chemistry, 16, 928–930

Hydrometer

G. J. Bouyoucos; The hydrometer as a new method for themechanical analysis of soils, Soil Science, 23 (5), 343–354

1918

1923

1927

PipetteH. M. Andreasen; The grinding of materials. theoretical andexperimental researches on particle-size distribution incident to thedisintegration process, Kolloid Beihefte, 27, 349–358

X-ray

A. L. Patterson; The Scherrer formula for X-ray particle size
1939
determination, Physics Review, 56, 978–982

Hydrometer“Divers”

S. Berg; Ingenior videnskab, Skrifter, 2, Danish Acad. Tech. Sci,
1940
Copenhagen

Coulter counter

W. H. Coulter; Means for counting particles suspended in a fluid,
US patent No. 2656508

Oct,
1953

Sedimentation J. F. Richardson and W. N. Zaki; Sedimentation and fluidisation.
and fluidization Part 1, Trans. Inst. Chem. Eng. 32, 35–53

1954

Centrifugal
sedimentation

J. K. Donoghue and W. Bostock; Technique for particle-size
analysis by centrifugal sedimentation, Transactions of the Institutionof Chem. Engineers, 33 (1), 72–75

1955

Laser

W. M. Farmer; Measurement of particle size, number density, andvelocity using a laser interferometer, Applied Optics, 11,2603–2612

1972

Laser
diffraction

J. Swithenbank, J. M. Beer, D. S. Taylor, D. Abbot and G.C. Mc
Creath; A laser diagnostic for the measurement of droplet and 1976
particle size distribution, AIAA J, 76–79

1928

12

E. Obata, H. Watanabe and N. Endo; Measurement of size and size
distribution of particles by fluidization, Journal of Chemical
Engineering of Japan, 15, 23–28

1982

E. Obata and H. Watanabe; Measurement of particle sizes by
fluidization, Encyclopedia of Fluid Mechanics, Vol. 4, 221–236

1986

E. Obata, H. Takahashi, M. Akiyoshi, K. Ando and H. Watanabe;
Measurement of particle size distribution by superficial velocity andpressure drops across liquid–fluidized beds, Kagaku Kougaku
Ronbunshu, 14, 103–106
Fluidization

1988

E. Obata, H. Takahashi, M. Akiyoshi and K. Ando; Themeasurement of the size distribution of irregular particles by
1990
liquid–solid fluidization in a high Reynolds number, Journal of theSociety of Powder Technology, 27, 301–307
E. Obata and K. Ando; Particle size measurements by fluidization:
From laminar flow region to the turbulent flow region,
Encyclopedia of Fluid Mechanics, Supplement 2, 169–189
E. Obata, M. Maruyama, Y. Ohira, M. Akiyoshi and K. Ando; The
simultaneous measurement of crystal sizes and mass in a fluidized

1993

1996

crystallizer, Journal of the Society of Powder Tech., 33, 456–461
Buoyancy

J. P. Bardet and J. Young; Grain-size analysis by buoyancy method,
1997
Geotechnical Testing Journal, 20, 481–485
E. Obata, Y. Ohira and M. Ohta; New measurement of particle size
distribution by a buoyancy weighing–bar method, Powder 2009
Technology, 196, 163–168
T. Motoi, Y. Ohira and E. Obata; Measurement of the floating
particle size distribution by buoyancy weighing–bar method,
Powder Technology, 201, 283–288

Buoyancy
weighing–bar

Y. Ohira, K. Furukawa, R. Tambun, M. Shimadzu and E. Obata;
Buoyancy weighing–bar method: A particle size distribution
measurement using new settling method,
Sedimentological Society of Japan, 69, 17–26

Journalofthe

R. Tambun, T. Motoi, M. Shimadzu, Y. Ohira and E. Obata; Size
distribution measurement of floating particles in the Allen region bya buoyancy weighing–bar method, Adv. Powder Tech., 22, 548–552
Y. Ohira, K. Nakano, R. Tambun, M. Shimadzu, M. Ohta and E.
Obata; Size distribution measurement of floating spherical particles
by the buoyancy weighing–bar method, Kagaku Kogaku

2010

2010

2011

2011

Ronbunshu, 37, 310–316

13Chapter 2
Graphical Analogy of Particle Size Distribution among Andreasen Pipette,
Sedimentation balance, Fluidization Curve and Buoyancy Weighing-bar Methods
2.1 Introduction
The particle size distribution (PSD) is absolutely essential to treating powders. The
instruments for measuring the PSD are sieving screens, microscopes, sedimentation particle
analyzers, and laser-scattering devices. In the sedimentation particle analyzers, there areseveral methods to measure the PSD, such as the Andreasen pipette method, fluidization
method, sedimentation balance method, buoyancy weighing-bar method, etc [4]. All thosemethods can determine the PSD by graphical method and have graphical analogy with eachother. One of analogy is cumulative undersize mass fraction obtained by Andreasen pipetteand fluidization curve methods, and the other is the cumulative oversize mass fraction by thebuoyancy weighing-bar and sedimentation balance methods. The Andreasen pipette method
determines the undersize fraction of the PSD by plot of cumulative mass undersize vs. settling
time. The fluidization curve method determines the undersize fraction of the PSD by plot ofpressure drop vs. superficial velocity. On the other hand, the buoyancy weighing-bar andsedimentation balance methods determine the oversize fractions of the PSDs by plot ofapparent mass of the weighing bar or cumulative mass oversize vs. settling time [1-6].
Theoretically, the buoyancy weighing-bar method (BWM) developed by us can beapplied not only to measure the settling particles, but also the floating particles with different
size [8]. In this paper, we focused to develop the BWM as a novel method in PSD
measurement.
2.2 Theory
The PSD can be classified as either a cumulative (R(x) or D(x)) or frequency (f(x))
function [7]. In this paper, all of the PSD signify the cumulative undersize D(x) or oversize
R(x) mass fraction. The settling velocity of a particle in a fluid is naturally the terminal
velocity in the laminar region.
2.2.1 Graphical analogy of PSD between the Andreasen pipette and fluidization curve
methods
2.2.1.1 Andreasen pipette method
The maximum and minimum diameters of particles in a homogeneous suspension are
symbolized as xmax and xmin, respectively. At a given time t from the beginning of
sedimentation, a particle x having the terminal velocity v(x) = h/t (which at the start were in
14

the top layer) have passed the point h. All of the particles having a velocity greater than h/t
have passed below the point h and approached the bottom. The initial concentration of thepowder is C0 and the initial density of the suspension is S0. At t = t, the concentration C of thepoint h and the density St are given by the following equations (cf. Figure 1.2 in Chapter 1): ρ  ρL  xmax
(2.1)
ρS 0  ρL  P
C0 
f ( x) dx
xminρPρSt  ρ L  ρP  ρL C x

xmin f ( x)dx

ρP

(2.2)

where L is the liquid density, P is the particle density, f(x) is the mass frequency of particle
xmax
h x 2  ρP  ρL  gsize x, ,f ( x) dx  1 , x is the particle size corresponding to: v( x)  
xmin
t
18 μ
where g is the gravitational acceleration, μL is the viscosity of the suspension liquid.
From Eqs. (2.1) and (2.2), the cumulative undersize mass fraction D(x) is:x
ρSt  ρ LCf ( x) dx  D( x)xminρ S 0  ρ L C0

(2.3)

The cumulative undersize mass fraction D(x) and the particle diameter are constructed as
Figure 2.1. Hence, the concentration of the point h have been the initial concentration C0from t = 0 to t = tmax.
2.2.1.2 Fluidization curve method
Obata et al. have reported five measurements for the PSD by fluidization in the laminar
to turbulent flow region, and one of them can be obtained by a graphical measurement using a
fluidization curve, i.e. the superficial velocity vs. pressure drop [1-5]. There are partially
fluidized and fixed beds in a fluidization system. The total pressure drop at a superficial
velocity u becomes P, which can be expressed as:P = (pressure drop in fluidized section) + (pressure drop on fixed bed).
The relationship between pressure drop and mass frequency of particles present in terms of
the fluidized particles of size xmin to x and the particles in the fixed bed of size x to xmax can be
expressed as [2]: ρ  ρ   x
ΔP  P L M
ρP Axmax  xmax f ( x )dx
 f ( x )dx 
o
x
x
umax
 x 
 min





u

2





(2.4)

where M0 and A are the total mass of particles in a fluidization bed and cross-sectional area ofthe bed. Figure 1.1 in Chapter 1 schematically illustrates the calculating method of the PSD
used in fluidization curve. The PSD measurement graphics used in fluidization curve method(Figure 1.1) is analogous to that of PSD measurement graphics by Andreasen pipette method(Figure 2.1).
2.2.2 Graphical analogy of PSD between the BWM and the sedimentation balancemethod15

2.2.2.1 BWM
Obata et al. have reported that the new measurement of the PSD by the BWM wasanalogous to the analysis of the sedimentation balance [4,6,8]. Figures 1.2 and 1.3 in Chapter
1 illustrate the schematic diagram of particle settling and schematic illustrate the calculating
method of the PSD used in the BWM, respectively. The line in t = 0 to tmax is theoreticallystraight.
2.2.2.2 Sedimentation balance method
The theory of BWM is the similar to that of the sedimentation balance method. There is
the pan for weighing the settling particles at the height h from a surface level. At the settling
time t, the weight of the pan is as following equation [7].
xmaxx v( x) tf ( x) dx  M 0
f ( x) dx
xxmin hwt  M 0 



(2.5)

where M0 is the total mass of the particles on the pan.
By differentiating with respect to time and multiplying by t,tx v( x) t
dw M0 f ( x) dx
x
min
dt
h

(2.6)

i.e.

 dw  (2.7)
wt  M 0 R( xt )   t   w0 wRt  wRt wt
 dt  The PSD graphical measurement used in sedimentation balance method (Figure 2.2) isextremely analogous to that of PSD graphical measurement by the BWM (Figure 1.3 in
Chapter 1). Naturally the line in t = 0 to tmax is straight.
2.3 Material and Method
Table 2.1 shows the particle size of the sample particles. The sample particles wereglass beads J–400 (density: 2.40  103 kg/m3) and JIS Test Powder 2, white fused alumina no.2
(density: 3.91  103 kg/m3) and no.3 (density: 3.93  103 kg/m3). The particle densities were
measured by the pycnometer.
Figure 2.3 schematically illustrates this experiment. The description of weighing bar isshown in Figure 2.4. The particle suspension was placed in a 1000 ml measuring cylinder(diameter: 65.0 mm, Sanplatec Co., Ltd). The weighing bar was hung from the analyticalbalance (GR-300: A&D Co., Ltd, minimum readout mass 0.1 mg) using a hanging wire. A
personal computer was connected to the analytical balance with RS–232C to record data
automatically. The weighing bar used in this experiment was aluminum slit cylinder (O.D.: 40
mm, I.D.: 30 mm, cylinder length: 210 mm, slit depth: 5.0 mm, slit length: 200 mm,
submerged length: 200 mm, density: 2.70  103 kg/m3).
The experimental apparatus was placed in a box to avoid external effects such as
airflow and temperature changes. A heater connected to the thermostat was placed in the box
16to maintain the desired temperature of 298 K.
The suspensions had a solid concentration of 10 kg/m3 (ca. 1 wt% in water). The
suspensions were mixed in measuring cylinder. After thoroughly stirring the suspension using
an agitator, the bar was set with the balance and was equipped with a personal computer. The
measuring data, which consist of time t and the corresponding mass of the bar GB, wererecorded on a personal computer. The measuring time was 2 hours and the data were collected
in 5–second intervals. After the measurements, the PSDs were calculated based on theabove–described theory. Additionally, the PSDs were compared to the results from thesedimentation balance, Andreasen pipette and laser diffraction/ scattering method (Microtrac
MT3000EX, Nikkiso Co., Ltd.).
2.4 Results and Discussion
Figure 2.5 shows the change with time in the apparent mass of the weighing bar GB
when the spherical glass beads J-400 was used. The apparent mass of the weighing bar
increased over time until all the particles settled below the lower end of the weighing bar, andthen the apparent mass of the weighing bar tended to become constant.
Figure 2.6 shows the PSD of spherical glass beads J–400. Based on the experimental
result, the BWM gave the close result with the data obtained by sedimentation balance andlaser diffraction/scattering methods, but not close to that measured by Andreasen pipettemethod. Hence, the BWM can measure the PSD of large particle up to 63 m in water.
Figure 2.7 and Figure 2.8 show the PSD of the white fused alumina no.2 and no.3,
respectively. At those Figures, the PSDs measured by the BWM were close to those measured
by Andreasen pipette and sedimentation balance methods. Figure 2.9 shows the PSD of theparticle mixture between no.2 and no.3 of white fused alumina by the BWM. The weightpercentage of white fused alumina no.2 and white fused alumina no.3 were 40% and 60%,
respectively. Based on the experimental result, the BWM can be used to measure the PSD of
two-component particles with different size and gave the similar result with the data obtainedby Andreasen pipette and sedimentation balance methods. As the particles of white fusedalumina were needle-shaped not spherical, the PSDs measured by laser diffraction/scattering
were slightly difficult from those measured by sedimentation particle analyzers.
As for the BWM, operability is easy, a personal error is small, and the price of thedevice is low.
2.5 Conclusions
We have investigated that there is the graphical analogy of PSD among Andreasen
pipette, sedimentation balance, fluidization curve and buoyancy weighing-bar methods. Using
the BWM, we measured the settling PSDs of glass beads J–400 and JIS Test Powders 2, whitefused alumina no.2 and no.3. The following conclusions were determined:171. The PSD of large particle up to 63 m can be measured in water using the BWM.
2. The BWM can measure the PSD of a needle-shaped particle for single and binary particles,
and the result data obtained comparable to that measured by Andreasen pipette andsedimentation balance methods.

18

Nomenclature
A
C
D(x)
f(x)

cross–sectional area of the bed, m2
solid concentration of suspension, kg/m3
undersize mass fraction of particle size x, –mass frequency of the particle size x, m-1

g
GBt
GRt

gravitational acceleration, m/s2
mass of weighing bar at t = t, kg
VB  B  wRt , kg

h
Mo
R(x)

submerged length of weighing bar, m
total particle mass, kg
oversize mass fraction of particle size x, –

t
u
v(x)
VB
w

time, ssuperficial velocity, m/s
sedimentation velocity of particle size x, m/s
submerged volume of weighing bar, m3

wB

mass of sediment, kg
buoyant mass of the submerged weighing bar in the suspension, kg

wRt

wB0 

x

particle size, mliquid viscosity, Pa · sliquid density, kg/m3
density of weighing bar, kg/m3
particle density, kg/m3

L
L
B
P
S

xmax
xmax
f ( x) dx or M 0
f ( x) dx, kg
xx

density of suspension, kg/m3

Subscripts
max
maximumminminimum
0
initial t = 0∞
infinity t = ∞19

Undersize fraction, D(x) [–]

1.0

1.0

c
co

0

xmax

Particle size, x [m]

x1

xi

xmin

0

tmax t1

ti

Settling time, t [s]

tmin

1
xmax
1
x1

ti 

18  L h p   L gxi2

1
 a  xi 

2

1
xi

1
xmin

Particle mass, w [kg]Oversize fraction, R(x) [–]Figure 2.1 Graphical determination of PSD in Andreasen pipette method.

1.0

0

xmax

x1

x2

Particle size, x [m]

w∞
wt
wRt
wR2

wR1
w0

xt xminT

tmax t1

t2

1
xmax
1
x1

t

Settling time, t [s]

ti 

1
18  L h a 
 p   L gxi2  xi 

tmin

2

1
x2
1
xt
1
xmin

Figure 2.2 Graphical determination of PSD in sedimentation balance method.

20

1
4

3

2

5

6

7

8
9

1. Personal computer 2. Analytical balance 3. RS –232C cable 4. Hanging wire
5. Weighing bar 6. Measuring glass cylinder 7. Thermal insulation vessel
8. Heating panel 9. Controller

Figure 2.3 Schematic diagram of experiment apparatus.

φ 40 mm
φ 30 mm

210 mm

200 mm5 mm

Figure 2.4 Weighing bar (Slit-cylinder type).

21

180.4
180.2
180
179.8
179.6
179.4
0

120

240

360

480

600

720

Time, t [s]Figure 2.5 Apparent mass of weighing bar as a function of time
(glass beads J-400).

Undersize mass fraction, D [-]Apparent mass, G Bt [x 10-3 kg]

180.6

1
0.8
0.6
Laser
buoyancy weighing bar
sedimentation balance
Andreasen pipette

0.4
0.2
0
0

10

20

30

40

50

60

70

80

90

100

Particle size, x [m]Figure 2.6 PSD of glass beads J-400.

22

Undersize mass fraction, D [-]

1
0.8
0.6

Andreasen pipettesedimentation balancebuoyancy weighing bar
laser

0.4
0.2
0
0

4

8

12

16

20

Particle size, x [m]Undersize mass fraction, D [-]Figure 2.7 PSD of white fused alumina no.2, JIS Test Powders 2.

1
0.8
0.6
0.4

Andreasen pipettesedimentation balancebuoyancy weighing bar
laser

0.2
0
0

4

8

12

16

20

24

28

32

Particle size, x [m]Figure 2.8 PSD of white fused alumina no.3, JIS Test Powders 2.

23

Undersize mass fraction, D [-]

1
0.8
0.6
0.4

Andreasen pipettesedimentation balancebuoyancy weighing bar
laser

0.2
0
0

4

8

12

16

20

24

28

32

Particle size, x [m]Figure 2.9 PSD of particle mixture of white fused alumina
no.2 and no.3 (40%:60%), JIS Test Powders no.2.

24

Table 2.1
Particle size of sample particles.

Sampleparticles

Particle size (m)94%
oversize

Median(80% or more)

50%
oversize3%
oversize

Upper
limit

White fusedalumina no.2

2.0 min–

4±0.5

11 max–White fusedalumina no.3

4.5 min–

8±0.6

20 max

–

–

< 45

–

–

63

J–400

25

Chapter 3
Measurement and Estimation of Settling Particle Size Distribution by the
Buoyancy Weighing-Bar Method and the Rosin-Rammler Distribution
3.1 Introduction
Water might often spring when a soil near the river or lake dug up by an excavator.
When spring water by digging contain soil particle, it is necessary to remove the suspended
solid by flocculation and/or sedimentation for environmental preservation. For removing thesuspended solid by flocculation and/or sedimentation, particle size distributions of soil andsuspended solid in spring water are important information. Measurement of the particle size
distribution (PSD) usually used the Andreasen pipette method [1], the hydrometer method [3],the centrifugal sedimentation method [5], the microscopy [16], the laser diffraction/scattering
method [4], and the Coulter counter method [15]. The operation of the Andreasen pipettemethod, the microscopy and the hydrometer method is difficult though the equipment of thoseis cheap. On the other hands, although other methods produce highly accurate results within ashorter time, the equipments are extremely expensive. Each method is difficult to measure the
PSD promptly on the construction field. It is necessary to develop the method of knowing the
PSD at a low price and easily.
We have reported that the buoyancy weighing bar method (BWM) capable ofmeasuring PSD of the settling particles [2,12,17]. The measurement result is almost equal to
the sedimentation balance method [13]. However, the particle size measured within 2 hours
was up to about 5 μm. To control the sedimentation process of suspended solid come from theconstruction field, it is necessary to determine the PSD of small particle which is less thanabout 5 μm. Particle size of less than about 5 μm has to measure for a long time or use theexpensive equipments. These are not suitable for the measurement on the construction field.
We thought that the PSD up to the small particle could be estimated by measuring data of the
BWM and the Rosin-Rammler distribution [14].
In this paper, we discuss the estimation validity of the PSD which used theRosin-Rammler distribution. For this estimation validity, we used the samples particles ofsuspended solids in spring water from a construction site near the Sorachi River at Furano,
Hokkaido and JIS Test Powders 1 [silica sand (class 3), fly ash (class 10), KANTO (Japanese)loam (class 11), calcium carbonate, heavy (class 17)]. Besides, we experimentallyinvestigated the PSD measurements of JIS Test Powders 1 (talc class 4, calcium carbonate(heavy) class 16, fly ash class 5), JIS Test Powders 2 (glass beads GBL 30, white fusedalumina No.4), and magnesite and soft–burned magnesia (China). The influences of materials
of weighing bar also were reported in this chapter.
3.2 Theory26

In this section, the outline is briefly described because the theory of the settling PSD by
the BWM has been explained in Chapter 1.2.1. The Rosin-Rammler distribution describes therelationship between particle size x and cumulative mass oversize R. Cumulative mass
oversize R can be expressed as a function of particle size x by
  x n R  exp    ,  xe  



(3.1)

where xe is an absolute size constant and n is the distribution constant. To determine theseparameters, Eq. (3.1) can be rewritten as x
1
ln ln   n ln   n ln x  n ln xe .
R xe 

(3.2)

Graphing lnln(1/R) versus ln x can determine distribution constant n from the slope.
Absolute size constant xe can be determined from the slope and intercept. The PSD can becalculated using Eq. (3.1), the distribution constant, and the absolute size constant.
3.3 Material and Method
The sample particles were silica sand class 3, talc class 4, calcium carbonate (heavy)class 16 and 17, fly ash class 5 and 10, KANTO (Japanese) loam class 11 (JIS Test Powders1), spherical soda lime–silicate glass beads GBL 30, white fused alumina No.4 (JIS Test
Powders 2), and magnesite (China, density: 3×103 kg/m3), soft–burned magnesia (China,
density: 2.4×103 kg/m3 ).
Spring water was from a construction site near the Sorachi River at Furano, Hokkaido.
Two liters of the suspension were separated into suspended solids and the dispersion liquid bya centrifugal separator (H-103N, Kokusan Co. Ltd.). The suspended solid concentration,
ignition loss, and solid density analyzed according to JIS K0102 [8], JIS A1226 [7] and JIS
A1202 [6] were 7.8 kg/m3, 0.041, and 2.71×103 kg/m3, respectively. NaHMP with aconcentration of 0.078 kg/m3 was used as a dispersant. The density and viscosity of thedispersion liquid added to NaHMP measured by the pycnometer [10] and Ubbelohde
viscometer [9] were 1.00×103 kg/m3 and 0.89 mPa.s, respectively. Table 3.1. shows theproperties of the sample particles [11].
Ion exchanged water and ethanol were used as liquid phase. The suspensions had asolid concentration of 10 kg/m3 (ca. 1 wt.%). Sodium pyrophosphate (NaPP) and sodium
hexametaphosphate (NaHMP), Kanto Chemical Co., Inc. were used as dispersant at using ion
exchanged water as dispersion liquid.
The schematic diagram of the experiment apparatus and detail view of weighing bar are
shown in Figures 2.3 and 2.4, respectively. The temperature during the experiment wasmaintained at 298 K. After thoroughly stirring the suspension using an agitator, the weighing
bar was set with the balance and was equipped with a personal computer. The measuring data,
which consist of time t and the corresponding apparent mass of the weighing bar GB, were27

recorded on a personal computer. The measuring time was one hour and the data werecollected in 5–second intervals. After the measurement, the PSD was calculated based ontheory and compare to the others method such as the laser diffraction/scattering (Microtrac
MT3000EX, Nikkiso Co., Ltd.), Coulter counter, sedimentation balance, Andreasen pipettemethods and the Rosin-Rammler distribution.
3.4. Results and discussion
Figure 3.1 shows the change with time in the apparent mass of the slit–cylinder when
the initial concentration of the JIS Test Powders 2, No.4 (white fused alumina) was 5.0 kg/m3.
The apparent mass of the slit–cylinder increased until all the particles settled below the lower
end of the slit–cylinder, and then the apparent mass of the slit–cylinder became constant. Thechange in the apparent mass was due to the change in the buoyant mass against theslit–cylinder as well as particle settling.
Figure 3.2 shows the influence of solid concentration on PSDs measurement by the
BWM when the JIS Test Powders 2, No.4 (white fused alumina) was used. The sodium
hexametaphosphate (NaHMP) was used as dispersant. When the initial concentration of the
JIS Test Powders 2, No.4 (white fused alumina) is 10 kg/m3 or less, there is no effect of initialconcentration on the PSD of the JIS Test Powders 2, No.4 (white fused alumina). When theinitial concentration is 50 kg/m3, the PSD of the JIS Test Powders 2, No.4 (white fusedalumina) could not be measured by the BWM. We found that the optimum solid concentration
was 10 kg/m3 or less. The PSD measured by the Andreasen pipette method is indicated by thefilled triangle plots in Figure 3.2. The PSDs measured by the BWM were close to thosemeasured by the Andreasen pipette method, but not close to those measured by the laser
diffraction/scattering method, because the particle shape of the JIS Test Powders 2, No.4
(white fused alumina) is needle–shaped.
The effect of weighing bar density on PSD was investigated experimentally. Figure 3.3
shows the PSDs of the non–spherical silica sand class 3, JIS Test Powders 1, when a bar
composed of aluminum, copper, acryl resin and SUS 304 were used. The material of theweighing bar did not influence the PSD of non–spherical silica sand class 3, JIS Test Powders
1. The PSDs measured by the BWM was close to those measured by the sedimentation
balance, Andreasen pipette and laser diffraction/scattering methods.
Figure 3.4 shows the PSDs of the non–spherical calcium carbonate (heavy) class 16,
JIS Test Powders 1, when a bar composed of aluminum, copper and SUS 304 were used. The
material of the weighing bar did not influence the PSD of non–spherical calcium carbonate(heavy) class 16, JIS Test Powders 1, and gave the similar result to those measured by
sedimentation balance, Andreasen pipette, microscope and laser diffraction/scattering
methods.
Figure 3.5 shows the PSDs of the JIS Test Powders 1, talc class 4. Ion exchanged waterand ethanol were used as dispersion liquid. Sodium pyrophosphate (NaPP, Kanto Chemical
28

Co., Inc.) with the concentration of 3.0 kg/m3 was used as dispersant at using ion exchanged
water as dispersion liquid. As shown in Figure 3.5, there is no influence of dispersion liquidon the PSDs. The PSD of the talc class 4 measured by the BWM agreed well to thosemeasured by sedimentation balance and Andreasen pipette methods, but not close to thosemeasured by laser diffraction/scattering method, because the particle shape of the JIS Test
Powders 1, talc class 4 is sheet–shaped. Hence, the BWM can measure the PSD of JIS Test
Powders 1, talc class 4.
Figures 3.6 and 3.7 show the PSDs of magnesite (China) and soft–burned magnesia(China), respectively. The PSD of the magnesite (China) and soft–burned magnesia (China)
measured by the BWM agreed well to those measured by laser diffraction/scattering method.
Hence, the BWM can measure the PSD of magnesite and soft–burned magnesia (China).
Figure 3.8 shows the PSDs of the JIS Test Powders 2, glass beads GBL 30 (sodalime-silicate). The PSD of the glass beads GBL 30 measured by the BWM agreed well tothose measured by Andreasen pipette, Coulter counter and laser diffraction/scattering methods.
Hence, the BWM can measure the PSD of glass beads GBL 30 (JIS Test Powders 2).
Figure 3.9 shows the PSDs of the JIS Test Powders 1, class 8 (KANTO (Japanese)loam). The PSD of the class 8 (KANTO (Japanese) loam) measured by the BWM agreed wellto those measured by sedimentation balance, Andreasen pipette and laser diffraction/scattering
methods. Hence, the BWM can measure the PSD of JIS Test Powders 1, class 8 (KANTO(Japanese) loam).
Figure 3.10 shows the PSDs of the JIS Test Powders 1, fly ash class 5. The sodium
pyrophosphate (NaPP) with the concentration of 3.0 kg/m 3 was used as dispersant. The PSDof the fly ash class 5 measured by the BWM agreed well to those measured by sedimentation
balance and laser diffraction/scattering methods. Hence, the BWM can measure the PSD of
JIS Test Powders 1, fly ash class 5.
3.4.1 Combination validity of Rosin-Rammler distribution
The PSDs of JIS Test Powders 1, class 3 (silica sand) measured by the BWM within 2hours was shown in Figure 3.3. The minimum particle size that could be measured in the
NaHMP solution is about 5 μm. To estimate particle sizes less than 5 μm, absolute size
constant xe and distribution constant n in Eq. (3.2) were calculated using that figure. Figure
3.11 shows the Rosin-Rammler plot. Because the slope and intercept are 0.883 and 2.24,
absolute size constant xe and distribution constant n are 12.6 μm and 0.883, respectively.
Figure 3.12 shows the PSDs of JIS Test Powders 1, class 3 (silica sand). The open circles anddashed line denote previous data [2] and the PSD calculated by Eq. (3.2), respectively. The
dashed line estimates cumulative mass oversize R at 1 μm is about 0.9. The filled circles in
Figure 3.12 show the standard data for the particle size by the sedimentation balance methodfrom Japan Industrial Standard, and the squares denote the PSD measured within 24 hours.
The PSD calculated by Eq. (3.2) is close to the standard data. Additionally, the PSD measured
29

within 24 hours agrees well with the dashed line. Hence, we conclude that the Rosin-Rammler
distribution and data measured within two hours by the BWM can estimate the PSD.
3.4.2 Estimation accuracy
The PSD can be calculated theoretically by Eq. (3.2) and two data points. However,more data is necessary due to measurement errors. The cumulative mass oversize for JIS Test
Powders 1, class 3 (silica sand) is up to about 0.6 within two hours. In this section, the lower
limit of cumulative mass oversize measured by the BWM and the estimation accuracy areexperimentally investigated.
Figure 3.13 shows the PSDs of JIS Test Powders 1, class 10 (fly ash) and class 11(KANTO (Japanese) loam) with a measurement time of two hours as well as the JIS standard
values and calculated values using Eq. (3.2). The values of the cumulative mass oversize are
up to about 0.25 (class 10) and about 0.15 (class 11), but the PSD less than about 5 μm remain
unknown within two hours. Using these data and Eq. (3.2), absolute size constant xe anddistribution constant n are calculated, respectively. The calculated line agrees well with the
JIS standard values.
Figure 3.14 shows the PSD of JIS Test Powders 1, class 17 (calcium carbonate, heavy)
as well as the calculation results from Eq. (3.2) and the JIS standard value. The value of
cumulative mass oversize is up to about 0.10 within two hours. In this case, the calculated line
does not agree with the JIS standard values. When the value of cumulative mass oversize
exceeds 0.15, the PSDs can be estimated by Eq. (3.2). Therefore, the PSD was measured
within six hours. The triangles and dashed line in Figure. 3.14 depict the PSD and thecalculated result using Eq. (3.2), respectively. The value of the cumulative mass oversize is up
to about 0.30 within six hours. The dashed line agrees well with the JIS standard values.
Hence, the results indicate that Eq. (3.2) can estimate the PSD when the cumulative mass
oversize exceeds 0.15.
3.4.3. Suspended solid in spring water
Figure 3.15 shows the PSD of suspended solids in spring water from a construction sitenear the Sorachi River at Furano, Hokkaido. The BWM measured the particle size of thesuspended solid from 5 to 71 μm within two hours. The values of cumulative mass oversize
are up to about 0.7. The dashed and solid lines in Figure 3.15 denote the calculated line by Eq.
(3.2) and the data from the laser diffraction/scattering method, respectively. Although theprinciples of the laser diffraction/scattering method and the BWM differ, the PSD calculated
by Eq. (3.2) provides results similar to those measured by the laser diffraction/scattering
method. Hence, the PSD of suspended solids can be estimated by the BWM and theRosin-Rammler distribution. Consequently, even without expensive laser diffraction/
scattering equipment, the PSD of fine particles at a construction site can be determined.
30

3.5 Conclusions
1. For measuring the PSD of the JIS Test Powders 2, No.4 (White fused alumina), thesuitable initial concentration is 10 kg/m 3 or less. The PSDs measured by the BWM wasclose to those measured by the Andreasen pipette method, but not close to those measured
by the laser diffraction/scattering method.
2. When a bar composed of aluminum, copper, acryl resin and SUS 304 were used tomeasure the PSD of non–spherical silica sand class 3, JIS Test Powders 1, the material of
the weighing bar did not influence the PSD. The PSDs measured by the BWM gave thesimilar result to those measured by the sedimentation balance, Andreasen pipette and laser
diffraction/scattering methods.
3. When a bar composed of aluminum, copper and SUS 304 were used to measure the PSD
of non–spherical calcium carbonate (heavy) class 16, JIS Test Powders 1, the material of
the weighing bar did not influence the PSD. The PSDs measured by the BWM gave thesimilar result to those measured by sedimentation balance, Andreasen pipette, microscopy
and laser diffraction/scattering methods.
4. BWM can measure the PSD of JIS Test Powders 1, talc class 4 used the ion exchanged
water and ethanol as dispersion liquid. The PSD measured by the BWM agreed well tothose measured by sedimentation balance and Andreasen pipette methods, but not close to
those measured by laser diffraction/scattering method.
5. BWM can measure the PSD of magnesite (China) and soft–burned magnesia (China). The
PSD measured by the BWM agreed well to those measured by laser diffraction/scattering
method.
6. BWM can measure the PSD of glass beads GBL 30 (JIS Test Powders 2) and gave theclose result to those measured by Andreasen pipette, Coulter counter and laser diffraction/
scattering methods.
7. BWM can measure the PSD of JIS Test Powders 1, class 8 (KANTO (Japanese) loam).
The PSD measured by the BWM agreed well to those measured by sedimentation balance,
Andreasen pipette and laser diffraction/scattering methods.
8. BWM can measure the PSD of JIS Test Powders 1, fly ash class 5. The PSD measured by
the BWM agreed well to those measured by sedimentation balance and laser diffraction/
scattering methods.
9. The PSDs of JIS Test Powders 1, class 3, class 10 and class 11 could be measured by the
BWM within two hours, and the Rosin-Rammler distribution can be estimated.
10. The PSD of JIS Test Powders 1, class 17 could not be estimated because the cumulative
mass oversize is up to about 0.1 within two hours.
11. The PSD of suspended solids in spring water obtained at a construction site could bemeasured by the BWM and its Rosin-Rammler distribution.

31

Nomenclature
D(x)
GBt
GRt

mass percentage undersize of particle size x, %mass of weighing bar at t = t, kg
VB B  WRt , kgndistribution constant, -Rtcumulative mass oversize, -xxetime, sparticle size, mabsolute size constant, m

32

Figure 3.1 Apparent mass of weighing bar as a function of time.

100

101

102

Figure 3.2 PSDs of JIS Test Powders 2, No.4 (white fused alumina).

33

Figure 3.3 PSDs of non–spherical silica sand class 3 (JIS Test Powders 1).

Figure 3.4 PSDs of the non–spherical calcium carbonate (heavy)class 16, JIS Test Powders 1.

34

100

101

102

Figure 3.5 PSDs of the JIS Test Powders 1, class 4 (talc).

Figure 3.6 PSDs of magnesite (China).

35

Figure 3.7 PSDs of soft–burned magnesia (China).

Figure 3.8 PSDs of spherical glass beads GBL 30 (JIS Test Powders 2).

36

100

101

102

Figure 3.9 PSDs of the JIS Test Powders 1, class 8 (KANTO (Japanese) loam).

Figure 3.10 PSDs of the JIS Test Powders 1, class 5 (fly ash).

37

Figure 3.11 Rosin-Rammler plot (JIS Test Powders 1, class 3 (silica sand)).

Rosin-Rammler

Figure 3.12 PSDs of JIS Test Powders 1, class 3 (silica sand).

38 BWM
Rosin-Rammler
 JIS

◇


BWM
Rosin-Rammler
JIS

Figure 3.13 PSDs of JIS Test Powders 1, class 10 (fly ash)and class 11 (KANTO (Japanese) loam).

BWM (2 h)Rosin-Rammler(2 h)
JIS
BWM (6 h)Rosin-Rammler(6 h)Figure 3.14 PSDs of JIS Test Powders 1, class 17 (calcium carbonate, heavy)

39

○

BWM
Rosin-Rammler
MT3000EX

Figure 3.15 PSDs of suspended solid in spring water.

40

Table 3.1
Properties of sample particles (granted by Japan Industrial Standard).
Median size[m]Particle density[kg/m3]

3.6 – 4.6  10-62700 – 2800

JIS Test Powders 1, silica sand class 3

6.6 – 8.6  10-6

2600 – 2700

JIS Test Powders 1, class 8 (KANTO(Japanese) loam)

6.6 – 8.6  10-62900 – 3100

JIS Test Powders 1, talc class 4

7.2 – 9.2  10-62700 – 2900

JIS Test Powders 1, fly ash class 5

13 – 17  102000 – 2300

-63900 – 4000

-62100 – 2500

Sample particle
JIS Test Powders 1, calcium carbonate (heavy)class 16

JIS Test Powders 2, white fused alumina No. 4
JIS Test Powders 2, glass beads GBL 30-614±1  10
30±1  10

41

Chapter 4
Graphical and Numerical Determinations of Mean Particle Sizeof Settling Particles by the Buoyancy Weighing-Bar Method
4.1 Introduction
The particle size distribution (PSD) and mean particle size are the most importantcharacteristics in particulate or powder technology. Sedimentation techniques to determine
PSD consist of the Andreasen pipette, manometoric pressure, sedimentation balance,
buoyancy weighing-bar and fluidization curve methods. The buoyancy weighing-bar andfluidization curve methods were developed by Obata et al and Stanly–Wood et al [1-4,8].
Tambun et al have reported that the PSDs among Andreasen pipette, sedimentation balance,
fluidization curve, and buoyancy weighing-bar methods have graphical similarities [10].
Because gravitation sedimentation methods use the settling velocity to determine particle size,
the time-consuming measurement process is inconvenient when determining PSDs. Except
for the Andreasen pipette method, sedimentation measurement methods determine the mean
size of all the particles in a relatively short time.
Recall how the minimum fluidization velocity is determined in a fluidization system.
Because the pressure drop of a fixed bed is proportional to the superficial velocity in alaminar flow region, the intersection point of the fluidization curve can determine theapparent minimum fluidization velocity of all the particles [1-3,6-9]. In the same manner, theincrease in the apparent mass must be proportional to the settling time over a short period
from the start until the maximum particles settle to the bottom of the sedimentation pan or
below the submerged height of the weighing bar. However, the aforementioned phenomena
have not been employed in particle size measurements.
This chapter proposes graphical and numerical determinations of the mean particle sizes
employing the buoyancy weighing-bar method (BWM) as a novel method to measure PSD.
The BWM with a slit-cylinder shaped weighing bar is used to determine the mean particle
size for a ternary mixture and multi-component particles. Similar to other methods to analyze
particle size, the BWM is accurate in an initially homogeneous suspension. Because the
BWM apparatus can be made by hand, it is also economical. Additionally, the BWM is easier
to be implemented compared to other settling particle analysis methods. The ability todetermine the maximum particle size and mean particle size in a short time is useful in the
powder industry.
4.2. Theory
4.2.1 Schematic diagram of particle settling42

Sedimentation methods to determine the PSD depend on Stokes law and assume theparticles in a suspension are spherical. For simplicity, the particles in this study are composedof three size and mass components xL, ML, xM, MM and xS, MS, respectively. Figure 4.1 is a
modified schematic diagram of particle settling with time, which was originally illustrated bySatone et al [5]. The volume of the submerged weighing bar is VB = Ah where A is the
cross-sectional area of the weighing bar and h is the length of the submerged weighing bar in
the suspension. The densities of the dispersion liquid and particles are denoted as L and P,
respectively. The initial solid concentration of the suspension is C0 [kg-solid/m3-suspension].
The Stokes velocity of a given particle size xi is expressed as v(xi) [11]xi2 ( ρP  ρL ) g hv( xi ) 

18 μL
ti

(4.1)

From Eq. (4.1), v(xL)tL = v(xM)tM = v(xS)tS = h.
At a settling time t  0 (Figure 4.1a), the initial density of the suspension (S0) isρ S 0  ρL 

C0
ρ  ρ 
ρP P L(4.2)

Because the maximum buoyancy mass of the submerged weighing bar is VB S0, the apparent
mass of the weighing bar in the initial suspension is(4.3)
GB0  VB  ρB  ρS0 
where B is the density of the weighing bar. The apparent mass of the weighing bar istransposed to G0 ≡ 0 at t = 0.
At 0 < t < tL (Figure 4.1b), the decreasing ratio of the buoyancy mass of the weighing bar
is constant. The decreasing buoyancy mass for t = t is
CV t
(4.4)
Gt  0 B v( xL ) M L  v( xM ) M M  v( xS ) M S 
M 0h
where M0 = ML + MM + MS.
At t = tL (Figure 4.1c), all the large particles have settled below the weighing bar, while
the medium and small particles have settled to v(xM)tL and v(xS)tL, respectively. The buoyancy
mass at t = tL is
CV t
GtL  0 B L v( xL ) M L  v( xM ) M M  v( xS ) M S 
(4.5)
M 0h
At tL< t < tM (Figure 4.1d), the decreasing ratio of the buoyancy mass of the weighing
bar is constant again. The decreasing buoyancy mass for tL < t < tM is smaller than that for 0 <t < tL. The decreasing buoyancy mass from t = tL to t = tM is
C0VB (t  t L )v( xM )M M  v( xS )M S 
(4.6)
M 0h
At t = tM (Figure 4.1e), all the medium particles have settled to the height h. Thus, thelarge and medium particles do not contribute between h = 0 and h = h. The buoyancy mass at t
= tM is43

GtM 

C0VB
t v( x )M t v( x )M t v( x )M 
M 0h L L L M M M M S S(4.7)

At t = tS (Figure 4.1f), the small particles have settled to height h. From time tS, thebuoyancy mass of the weighing bar is constant. Therefore, the transposed buoyancy mass ismaximized.
CV(4.8)
GtS  G  0 B t L v( xL ) M L  t M v( xM ) M M  t S v( xS ) M S 
M 0hUsing Eqs. (4.4)–(4.8) and Figure 4.1, Figure 4.2 illustrates the decreasing buoyancy mass
ratio of the weighing bar as a function of settling time, and yields the following relations
M C '0M L AB M M BC ',and S (4.9)


M 0 A0M 0 A0 M 0 A04.2.2 Definition of maximum particle size and velocity mean particle size of all theparticles
The largest and the smallest particle sizes are xn = xmax and x1 = xmin, respectively. A
given particle size xi corresponds to time ti and the oversize fraction R(xi) is obtained by theintersection Gi. The mean size of all the particles and the apparent mean size of particles fromx1 to xi are expressed as x0 and x i , respectively. The mean size of the particles from xi to xn
is x i , and is above the line. In contrast, the mean size of the particles from x1 to xi is x i , and is
below the line. The apparent minimum fluidization velocity ( u mf ) of multi-component
particles has been determined by measuring the pressure drop through the particles in the
fixed bed state. The procedure for determining the minimum fluidization velocity ofmulti-component particles was proposed by Obata et al. (1982) and Rincon et al. (1994). It
was the following equation:

 n Mi 
u mf   i 1 M 0 u mfi 

1

 n

 M i  M 0 
 i 1(4.10)As the minimum fluidization velocity ( u mfi ) of single particle is expressed by the KozenyCarman equation, Eq. (4.10) is rewritten by1

 n Mi  2

x0  2 
 i 1 M 0 xi 

(4.11)

where x0 refers to the apparent mean particle size of all the particles. Naturally, x0
corresponds to the apparent minimum fluidization velocity of multi-component particles. As
described above, the curve of measuring mass vs. settling time by the buoyancy weighing-bar
method is same as the weighing characteristics in ideal particle settling.
In Figure 4.2, the largest particles ( x L ) settle to the bottom height (h) of the weighing bar
at t L . As the large, medium and small particles have settled to v( xL )t L , v( xM )t L andv( xS )t L , respectively, the decreasing buoyancy mass is constant. Therefore, the line Aδ must44

be a straight line and the maximum particle size must bexL 

18 μ L h(4.12)g ( ρP  ρL )t L

As the angle of the straight line Aδ corresponds to the mass ratio of ML, MM and MS, theintersection t 0 = t L of the extrapolated straight line Aδ and the total mass ML + MM + MS =M0 must be compatible with the mean particle size of all the particles ( x0 = x L ). This mean
particle size defines the velocity mean particle size in terms of the Stokes velocity.
x02 ( ρP  ρL ) gh v ( x0 )  v ( x L ) 
18 μL
tL

(4.13)

Hence, the velocity mean particle size is not the median particle size based on mass butthat is extremely useful in the powder industry, because the velocity mean particle size which
is supplied by the sedimentation techniques can be measurable in a few minutes.
The intersection t M of the extrapolated straight line δδ’ and the total mass corresponds
to the velocity mean particle size of MM and MS. Therefore,
h

tM

2

 v( x M ) x M ( ρP  ρL ) g
18 μL(4.14)

4.2.3 Velocity mean particle sizes of all the particles and from a given particle size to aminimum size
In the BWM, the initial concentration of the suspension, the volume, and the height of
the weighing bar do not have principled relations when determining the PSD. However, as thelargest particles settle to the bottom height of the weighing bar, the decreasing buoyancy mass
ratio is constant at a maximum value. Figure 4.2 depicts the relationship between thebuoyancy mass ratio and settling time.
At 0 < t < tL, the maximal decreasing buoyancy mass ratio is 1.0. Because the large
particles have settled to the bottom height of the weighing bar at t = tL, the decreasing
buoyancy mass ratio depends on MM and MS from this point. From Eq. (4.5), the decreasing
buoyancy mass ratio at t = tL corresponds to GtL = δ. The intersection t 0  t L  t L of theextrapolated straight line Aδ corresponds to the velocity mean size of all the particles x0 .
From Eq. (4.1), x0  x L 

18 μ L hg ( ρP  ρL )t L

(4.15)

Hence, the velocity mean size of all the particles x0 is same as the velocity mean particle size
from the maximum size to the minimum size x L . Further details are discussed later.
At t = tM, the medium particles have settled to the bottom height of the weighing bar,and the decreasing buoyancy mass ratio corresponds to GtM = δ’. The intersection D(xM) =(MM + MS)/M0 of the intercept at the point δ’ refers to the mass fraction undersize D(xM). In
45

contrast, the intersection t M > tM of the extrapolated straight line δδ’ corresponds to thevelocity mean particle size of MM and MS. Therefore,xM 

18 μL h(4.16)g ( ρP  ρL )t M

At t = tS = t S , all the particles have settled to the bottom height of the weighing bar.
The small particle size is same as the velocity mean particle size from the small particle size
to the minimum size,xS  x S 

18 μ L h(4.17)g ( ρP  ρL )t S

The mass ratio of all the small particles corresponds to line segment C’0, and the point C’refers to MS/M0 and undersize mass fraction D(xS).
At t = tL, the mass ratio of the small particles, which have settled to v(xS)tL at tL,
corresponds to line segment C’D’. Similarly the mass ratio of the medium and small particles
corresponds to segment BD. Therefore, the mass ratios of large, medium, and small particles
at tL correspond to AB, BC, and CD = C’D’, respectively. Because triangle AD is similar to
A0tL, the Stokes velocity of the velocity mean size of all particles is expressed byh x ( ρ  ρ )gv( x0 )   0 P L . The particle sizes of large and medium particles are naturally
18 μL
tLobtained by x L 

18 μ L hg ( ρP  ρL )t L

and x M 

18 μ L hg ( ρP  ρL )t M

, respectively.

4.2.4 Velocity mean particle size in the BWM
Obata et al and Stanly–Wood et al have developed PSD measurements of dispersed
particles in a fluid using a fluidization curve and the buoyancy of a weighing bar [1-4,8].
Tambun et al have reported that the Andreasen pipette, sedimentation balance, fluidization
curve, and buoyancy weighing-bar methods produce graphically analogous PSDs [10].
Figure 4.1 schematically depicts particle settling of a ternary mixture. If the raw
particles are comprised of three components with particle sizes xL, xM and xS, the apparent
mass of each type of particle in the volume of the submerged weighing bar is GL, GM, and GS,
respectively. Figure 4.3 shows the relationships between the buoyancy mass of eachcomponent or all the particles and settling time. At incipient time, t = 0, it is assumed that theapparent buoyancy mass of the weighing bar is zero. At t = tL, all large particles xL settle atheight h. When all small particles xS settle at t = tS, the apparent buoyancy mass of theweighing bar is GtS = G∞ = GL + GM + GS. Thus, the following equations hold.
δ = α + β + γ = GtLδ’ = α’ + β’ + γ = GtM(4.18)(4.19)
46

δ’’ = α’’ + β’ + γ = GS + GM + GL = GtS = G∞
(4.20)The velocity mean particle size x 0 of the ternary mixture refers to the intersection of theextrapolated straight line 0 because all the particles are settling within settling time tL.
Similarly, the velocity mean particle size x M of a binary mixture refers to the intersection of
the extrapolated straight line ’ because large particles settle below the weighing bar.
According to Figure 4.3,tan∠δ’GLγ = tan∠β’0tM + tan∠α’0tM(4.21)The velocity mean particle size of small particles ( x S ) is the intersection of the extrapolated
straight line ”’ because small particles are only present at the level of the weighing bar after
time tM. The oversize mass fraction R(x) = 1 – D(x) of particle sizes xL, xM, and xS is
determined in Figure 4.3. The relationship between settling time t and particle size x is
expressed by Stokes equation as Eq. (4.1). The velocity mean size of all particles ( x0 ) and thevelocity mean particle size from the medium particle size xM to small particle size xS ( x M )are derived by the following equations. From Eq.(4.21),
M
t 
t 
t 
α  M S  L  , β  M M  L  , γ  M L  L  , and C0VB i  ΔGi ,M0
 tM 
 tL 
 tS 
h
t 
h 
t 
 t   h G    GtL     ΔGS  L   ΔGM  L   ΔGS  L    tL   tM 
 t L   t L  t0 
 tS 
= ΔGS v( xS )  ΔGM v( xM )  ΔGL v( xL )(4.22)

where G∞ = GS + GM + GL. Therefore,
1

1

2
2
 M S xS2 M M x M
 ΔGS xS2 ΔGM xM
M L xL2  2
ΔGL xL2  2x0     G

M
M
M
G
G



0
0
0





(4.23)The velocity mean particle size from the medium particle size xM to small particle size xS is
derived by Eq. (4.24).
1

2
 ΔGS xS2 ΔGM x M
2
x M  

G 
 G

(4.24)

4.2.5 Particle size distributions and velocity mean particle sizes in multi-component
mixtures
Figure 4.4 depicts the relationship between the buoyancy mass and settling time inmulti-component mixtures. The largest and the smallest particle sizes, xmax = xn and xmin = x1,
are described at time tmax and tmin, respectively. A given particle size xi corresponds to time ti
and the oversize fraction R(xi) is obtained by the intersection Gi. The velocity mean size of all
the particles and the apparent mean size of particles from xmin to xi are expressed as x0 and x i ,respectively. The velocity mean size of the particles from xi to xmax = xn is x i , and is above theline. In contrast, the velocity mean size of the particles from xmin to xi is x i , and is below theline. From sedimentation balance theory,47

Gti  G 

xmax

xi

 dG f ( x )dx B  t i  0Gi Gi Gti
 dt  ti(4.25)

where f(x) is a function that defines the frequency distribution of the particles. The cumulative
oversize mass fractions R(xi) is
R( xi )  

xmax

xi

f ( x )dxGi
1 D( xi )G(4.26)The PSD of all the particles is shown in Figure 4.4. The maximum size of the particles isdefined by tmax.
Velocity mean particle sizes x0 and xi of all the particles as well as particles from a given
size xi to a maximum size xmax have been defined by the fluidization curve method for the
PSD [8]. Meanwhile, velocity mean particle sizes x0 and x i can be obtained by the BWM.
Figures 4.3 and 4.4 clearly show thatt
ΔGi max n
G Gt max
ti
ΔG


 i
t0
t max i 1 t max
i 1 t in(4.27)

wherenΔG Gi 1iHence, Gi is the buoyancy difference mass of particles xi. Multiplying all the terms by theheight of the weighing bar h yieldsnG v( x0 )  Gt max v( xmax )  ΔGi v( xi )i 1(4.28)

wherev ( x0 ) hhh, v( xmax ) and v( xi ) 
t0
ti
t max

At t = ti,G  Gi i ΔGiti
i 1 t i(4.29)

whereiG  Gi ΔGii 1

Therefore,1

 n ΔGi xi2  2
x0   

G
i

1



1

 i ΔG x 2  2
x i    i i 
 i 1 G  n ΔGi  G 
 i 1(4.30) i ΔGi  G  Gi  i 1(4.31)

Additionally, the velocity mean particle size of all particles x0 must be the summation, which48

is large mean size x i determined by adding small mean size x i
 G 
G 
(4.32)x0  1 i  x i   i  xi
 G 
 G 
Figure 4.5 shows all the mean sizes. It is obvious that all the equations in the BWM are

realized in the sedimentation balance and manometric pressure methods.
4.3 Materials and experiment4.3.1 Sample particles
The sample particles were spherical glass beads [GBL 60, GBL 40 and GBL 30, JIS
Test Powders 2 (soda lime-silicate glass)]. According to pycnometer measurements, theparticle densities of the sample materials were 2.30 – 2.37  103 kg/m3. Table 4.1 shows theparticle size of 10%, 50% and 90% oversize measured by a Coulter counter (Japan Industrial
Standard). The particle sizes with 50% oversize of glass beads of GBL 30, GBL 40 and GBL
60 were 30 µm, 41 µm and 59 µm, respectively.
4.3.2 Experimental apparatus
The full details of experiment apparatus and weighing bar can be seen in Figures 2.3
and 2.4, respectively. Briefly, the particle suspension was placed in a 1000 ml graduatedcylinder (diameter: 65.0 mm). The weighing bar was hung from an analytical balance(GR-300: A&D Co., Ltd., minimum readout mass 0.1 mg) using a hanging wire. A personal
computer was connected to the analytical balance with RS-232C recorded data automatically.
The weighing bar was an aluminum slit cylinder (O.D.: 40 mm, I.D.: 30 mm, cylinder length:
210 mm, slit depth: 5.0 mm, slit length: 200 mm, submerged length: 200 mm, density:
2.70  103 kg/m3).
The experimental apparatus was placed in a box to avoid external effects such as
airflow and temperature changes. A heater connected to the thermostat was placed in the box
to maintain the desired temperature of 298 K.
4.3.3. Experimental conditions and methods
Glycerol (Kanto Chemical Co., Ltd.) was used as an improver. The liquid phase was a
glycerol solution (concentration: 40 wt%). The density and viscosity measured by thepycnometer and the Ubbelohde viscometer were 1.064 – 1.086  103 kg/m3 and 2.32 –2.91  103 Pa･s, respectively. The suspensions had a solid concentration of 10 kg/m 3 (ca. 1
wt.% in glycerol solution). There is no effect of a solid concentration on the PSD when theconcentrations are 10 kg/m3 or less [12]. When a ternary mixture was used, the masspercentage of GBL 60, GBL 40 and GBL 30 were 30wt%, 40wt% and 30wt%, respectively.
49

After thoroughly stirring the suspension using an agitator, the weighing bar was set
with a balance, which was hooked to a personal computer. The measured data, which
consisted of time t and the corresponding apparent mass of the weighing bar GB, wererecorded on a personal computer in 5-s intervals for 1 h. Afterwards, the velocity mean
particle size were analyzed based on theory. Additionally, the velocity mean particle sizes
were compared to the calculated results. The calculation mode means that each type of glass
bead (GBL 60, GBL 40 and GBL 30) was measured separately by the laser
diffraction/scattering method (Microtrac MT3000EX, Nikkiso Co., Ltd.), and then determinedbased on the aforementioned theory.
4.4. Results and discussion
Figure 4.6 shows the change with time in the apparent mass of the weighing bar (GB)when spherical glass GBL 30 beads were used. The apparent mass of the weighing bar
increased over time until all the particles settled below the weighing bar. Because thetendency of the apparent mass to increase changed at 620 s, the maximum particle size oftested sample was1(18)(0.0026)(0.2)  2
 = 35.2 µm.
xmax  
 (9.81)(2307  1064)(620)

The apparent mass of the weighing bar became constant at 1300 s. The minimum particle sizeof tested sample was12(18)(0.0026)(0.2)
 = 24.3 µm.
xmin  
 (9.81)(2307  1064)(1300) As shown in Table 4.1, particle sizes with 10% and 90% oversize were 34 µm and 26 µm,
respectively, which are very similar. Thus, the change in the apparent mass of the weighing
bar over time can be used to calculate the particle size range. In Figure 4.6, the mean size ofglass beads GBL 30 corresponded to Stokes equation. For example, from t 0 = 840 s,
1(18)(0.0026)(0.2)  2
 = 30.2 µm,
x0  
 (9.81)(2307  1064)(840)

whereas the particle size with 50% oversize by the laser diffraction/scattering method was
30.1 µm. Similar analyses were performed for GBL 40 and GBL 60. In these cases, the range
of Reynolds number is 0.0015-0.047.
Figure 4.7 shows the PSDs of spherical glass beads of GBL 30, GBL 40 and GBL 60
measured by the BWM. The lines indicate data from the laser diffraction/scattering method.
Table 4.1 lists the 50% oversize for particles measured by the BWM and the laser
diffraction/scattering method. The BWM, laser diffraction/scattering method, and Coultercounter (Japan Industrial Standard) produced similar particle sizes with 50% oversize
50measured. Thus, the BWM and laser diffraction/scattering method provide similar PSDs.
Figure 4.8 shows the change in the apparent mass of the weighing bar (GB) over time
when a ternary spherical particle mixture of glass beads (GBL 60, GBL 40 and GBL 30) wasused. The apparent mass of the weighing bar increased over time until all the particles settled
below the weighing bar, and the apparent mass of the weighing bar became constant. Because
the tendency of the apparent mass of the weighing bar to increase changed at 200 s, themaximum particle size of sample mixture was 65.0 µm. The apparent mass of the weighing
bar became constant at 1300 s. The minimum particle size of sample mixture was 25.5 µm. As
shown in Table 4.1, the particle size with 10% oversize for GBL 60 was 63 µm, while theparticle size with 90% oversize for GBL 30 was 26 µm. Because these oversize values are
similar, it is concluded that the change in the apparent mass of the weighing bar with time canbe used to calculate the particle size range of the sample mixture.
Figure 4.9 shows the PSD of a ternary spherical particle mixture of glass beads (GBL
60, GBL 40 and GBL 30). The dashed and solid lines are the calculated results by multiplying
each mass percentage by the single GBLs and that from the laser diffraction/scattering method,respectively. The data from the BWM is close to the dashed line, confirming that the BWM
can measure the PSD of a ternary mixture. Although the reason is currently unclear, the PSD
measured by the laser diffraction/scattering method is inconsistent with the calculated result
for a ternary particle mixture.
In Figure 4.9, the velocity mean particle size x0 of the ternary mixture was 44.9 µm
because t 0 = 420 s. Once GBL 60 settled, the velocity mean particle size of GBL 40 and GBL
30 can be calculated. Likewise, the velocity mean particle size of GBL 30 can be calculated
after GBL 40 has settled. Table 4.2 shows the velocity mean particle sizes measured by the
BWM and those calculated by results of single particles. Equation (4.30) was used to
calculate the results from both the BWM and laser diffraction/scattering data. As described inTable 4.1, the results were calculated by multiplying each particle size with 50% oversize by
the single GBLs. Based on the experimental results, the BWM can measure the velocity mean
particle size of a ternary spherical particle mixture and provides results similar to thecalculated results. Hence, it is concluded that the BWM can determine the velocity mean
particle size of all the particles in a ternary mixture within 200 s.
4.5 Conclusions
The PSD and velocity mean particle size can be determined graphically by the BWM.
The following conclusions are determined:
1. The velocity mean size of all particles ( x0 ) and the velocity mean size of particles from x1
to xi ( x i ) by the BWM are determined by1

 n ΔGi xi2  2
x0   

G
i

1




 n

 ΔGi  G 
 i 1

51

1

 i ΔG x 2  2
x i    i i 
 i 1 G  i ΔGi  G  Gi  i 1

where Gi is the buoyancy difference mass of particles xi.
2. The maximum particle size can be defined by tmax.
3. The velocity mean sizes of particles ( x i ) can be recognized graphically by the BWM.
4. The PSD and mean particle size for a ternary spherical particle mixture of glass beads(GBL 60, GBL 40 and GBL 30) can be measured by the BWM because the BWM
measured and calculated results are similar.

52

Nomenclature
A
C0
D(x)
f(x)

cross-sectional area of the weighing bar, m2
initial solid concentration of the suspension, kg/m3
undersize mass fraction of particle size x, –

g
G
Gti
G∞
h
M

mass frequency of particle size x, m1
gravitational acceleration, m/s2
apparent buoyant mass of the weighing bar, kg
apparent mass of the weighing bar at t = ti, kg
GL + GM + GS, kg
submerged length of the weighing bar, mparticle mass, kg

M0

ML + MM + MS, kg

Re

Reynolds number ( 

R(x)
t
u
v(x)
VB

oversize mass fraction of particle size x, –time, ssuperficial velocity, m/s
settling velocity of particle size x, m/s

x
x0

xi
xi
ΔG
μL
ρB
ρL
ρP
ρS0

xv ( x) ρ L), –μL

submerged volume of the weighing bar, m3
particle size, mvelocity mean particle size of all particles, mvelocity mean particle size from xi to xmax, mvelocity mean particle size from xmin to xi, mbuoyancy difference mass of the particles, kg
liquid viscosity, Pa·s
density of the weighing bar, kg/m3
liquid density, kg/m3
particle density, kg/m3
initial density of the suspension, kg/m3

Subscripts
1
i
L
M
max
mf

first, minimum
i-th
large
mediummaximum
minimum fluidizationminnminimumn-th, maximum
53

S
∞

small
total

54

G0≡0

GtL

GtM

GtS = G∞

h=0

xL

xS

xM

h=h

(a) t = 0(b) t < tL(c) t = tL(e) t = tM(d) t < tM(f) t = tS

Figure 4.1 Schematic diagram of the change in ideal particle settling over time.

Mass ratio [–]t0  M L  M M  M S  (a)1t L M0
 A

D( xM ) 

M

M

M

M

S

0

(c)

B

0

C
tL  M M  M S DtM  M



(b)



MS
C’
M0
tM  M S  D’t S  M 0 

(c)

(d)

(e)

δ

CD = C’D’

D( xS ) 



(e)



0

0

(f)

δ'
tL

t0  t L

tM

tM

δ" (f)tS  t S

Settling time, t [s]Figure 4.2 Mass ratio vs. settling time in ideal particle settling.

55

Apparent mass, GB [kg]Oversize fraction, R(x) [–]

1.0

GL M
G
GL
G

0

xL

G∞

δ"
δ'

GL+M
GtL
GM
GL
GS

x0 x MxM xS  x S

0Particle size, x [m]MS+M+L

δ

β'

β'

γ

γ

γ

β

α'

α
0

tL

(xL ) 1

α"

t0

tM

ti 

18 μL h1a
ρp  ρL gxi2  xi tM

Settling time, t [s](x0 ) 1

MM
ML
MS

tS

2

(xM ) 1
(x M ) 1(xS ) 1

Apparent mass, GB [kg]Oversize fraction, R(x) [–]Figure 4.3 Determination of the particle size distribution and velocity mean
particle size of a ternary mixture using the buoyancy weighing bar method.

1.0

Gi

Gti

Gi
Gtmax

G

0

G∞

xmax

x0 x i
xi
xmin

0
0

Particle size, x [m]

1

xmax
1

tmax

ti

t0

ti

Settling time, t [s]ti 

18 μL hρ  ρ gx
p

L

2
i

1
a  
 xi 

tmin

2

x0
1

xi

1

xi
1

xmin

Figure 4.4 Determination of the particle size distribution and velocity mean particle
size of multi-component particles using the buoyancy weighing bar method.

56

G∞x

Frequency, Gi [m–1]

f (x)

0

0

xi

xi

xminx1

x0 xi

xmaxxn

Particle size, x [m]Apparent mass, GB – GB0 [x 10-3 kg]

Figure 4.5 Relationship between the mean particle size and mass frequency curve.

0.5

xmin

x0

xmax

GB∞–GB0

0.4
0.3
0.2
0.1
0
0

250

500

tmax

750

t0

1000

tmin

1250

1500

Time, t [s]Figure 4.6 Apparent mass of the weighing bar as a function of timewith glass beads (GBL 30, JIS Test Powders 2).

57

Undersize mass fraction, D [-]

1
0.8
0.6
BWM (GBL 30)
laser (GBL 30)
BWM (GBL 40)
laser (GBL 40)
BWM (GBL 60)
laser (GBL 60)

0.4
0.2
0
0

10

20

30

40

50

60

70

80

90

Particle size, x [m]Figure 4.7 Particle size distributions of glass beads (GBL 30, GBL 40
and GBL 60, JIS Test Powders 2).

Apparent mass, GB – GB0 [x 10-3 kg]

0.5

x0

GB∞–GB0 xmax

xM

xS

tM

tS

xmin

0.4
0.3
0.2
0.1
0
0

tmax

300

t0

600

900

1200

tmin

1500

Time, t [s]Figure 4.8 Apparent mass of the weighing bar as a function of time with theternary mixture (glass beads GBL 60, GBL 40 and GBL 30, JIS Test Powders 2).

58Undersize mass fraction, D [-]

1

0.8

0.6

0.4
BWM
calculated by BWM`s data
laser

0.2

0
0

20

40

60

80

100

Particle size, x [m]Figure 4.9 Particle size distributions of a ternary mixture of glass beads(GBL 60, GBL 40 and GBL 30, JIS Test Powders 2).

59

Table 4.1 Particle size of the sample particles.
PSD by Coulter counter (Japan Industrial Standard)[µm]
90% oversize

50% oversize

10% oversize

50% oversize
by BWM[µm]

GBL 3026 min30±1.034 max30.2

30.1

GBL 4037 min41±1.045 max

41.6

41.3

GBL 6055 min59±1.063 max57.8

58.0

Sampleparticles

50% oversize
by laser[µm]

Table 4.2 Mean particle size of a ternary mixture of glass beads GBL 60, GBL 40, and GBL 30

Composition

Mean particle sizegraphically by the BWM

calculated by BWM’s
datacalculated by laser’sdata

GBL 60, GBL 40,
GBL 30t0 = 420 s

x0 = 44.9 mx0 = 44.3 mx0 = 44.3 m

GBL 40, GBL 30

t M = 610 s

xM = 37.2 m

xM = 37.1 m

xM = 36.9 m

GBL 30t S = 840 s

xS = 31.7 m

xS = 30.2 m

xS = 30.1 m

60

Chapter 5
Measurement of the Floating Particle Size Distribution
by the Buoyancy Weighing–Bar Method
5.1 Introduction
Buoyancy weighing-bar method (BWM) is a settling method capable of measuring
particle size distribution (PSD) [1]. The accuracy of this method is similar to that of thesedimentation balance method or the Andreasen pipette method. Additionally, the material of
the weighing bar does not affect the results, and a particle size of 5 μm in water can bemeasured in two hours. Furthermore, the PSD of large particles in a viscous liquid can bemeasured using the BWM. We have reported PSD measurements of the floating particles by
the BWM [6,8,9]. The ability to measure the PSDs of floating solid particles suggests that thetheory for the size distribution measurement of the settling particle can be applied to floating
particles. The precision of the PSD is comparable to that of the laser diffraction/ scattering
method, which is a representative high precision method.
Becker et al. have reported sedimentation length and pan size affect the PSD using thesedimentation balance and settling particles [2]. Thus, we need to evaluate the influence ofparticle migration lengths as well as the size of the weighing bar and vessel on the BWM. In
this chapter, we reported the influences of the sizes and shapes of weighing bar and vessel on
the size distribution measurements of floating particles by the BWM. Also we experimentallyinvestigate the PSD measurement of hollow glass beads K1 (Sumitomo 3M), hollow glass
beads S60HS (Sumitomo 3M) and the glass beads GBL 100 (JIS Test Powders 2) by the
BWM.
5.2 Theory
In this section, the outline is briefly described because the theory of floating PSD
measured by the BWM has been explained in Chapter 1.2.2. Equation (5.1) shows thebuoyant mass balance for weighing bar in the suspension.

W0  W  (W0  W ) 

xmax

xi

f ( x)dx (W0  W ) 

xi

xmin

v( x)tf ( x)dx ,
h

(5.1)

where W is buoyant mass, h is length of the weighing bar, v(x) is the floating velocity of theparticle and f(x) is the mass frequency of the particle size x. The density of the suspension ρS
and the apparent mass of the weighing bar G in the suspension are given by the following
equations:
G  VB B  W  VB ( B  S ) ,(5.2)

where ρS is the suspension density, ρB is the weighing bar density in suspension and VB is the61weighing bar volume. Differentiate Eqs. (5.1) and (5.2) with respect to the time t, we obtain: dG 
(5.3)
G  GR  t .
 dt 
Value of GR calculates from tangent line based on Eq. (5.3). The cumulative mass oversize isxmax
G  GB0(5.4)
R( x)  
f ( x) dx  Rt
 1  D( x )xGB  GB 0

where D is cumulative mass undersize.
At Re < 6, particle size x is given by the following equation using Stokes formulamodified the voidage function:x

1 18 μL v( x) F (ε )φ g ( ρL  ρP )(5.5)

where, φ is Wadell’s shape factor, g is the gravitational acceleration, μL is the viscosity ofthe dispersion liquid contained the dispersant, ε is voidage, and F(ε) is the voidage function.
Wadell’s shape factor for a cylindrical particle φ is calculated using Eq. (5.6) [7] assurface area of a sphere with the same volume as the particle(5.6)
φ=
surface area of the particle
The voidage function is calculated by the Richardson and Zaki correlation [3].
F (ε)  ε 4.65(5.7)

The floating velocity of the particles v(x) can be calculated by Eq. (5.8):v( x) 

h
t

(5.8)

where h is the submerged length of the weighing bar and t is the floating time.
5.3 Materials and Methods
5.3.1 PSD measurement of glass beads GBL 100 (JIS Test Powders 2) and hollow glass
beads K1 and S60HS (Sumitomo 3M) by the BWM
The details of the experimental apparatus are shown in Figure 5.1. The samples
particles are glass beads GBL 100 JIS Test Powders 2 (density: 2.30×10 3 kg/m3), hollow glass
beads K1 Sumitomo 3M (density: 0.125×103 kg/m3) and hollow glass beads S60HS
Sumitomo 3M (density: 0.600×103 kg/m3). The particles suspensions were placed in the 1000
ml graduated cylinder (diameter: 65 mm).
Figure 5.2 schematically depicts the weighing bars. For sample particle glass beads
GBL 100 JIS Test Powders 2, the titanium weighing bar (diameter: 10 mm, length: 250 mm,
density: 4.20×103 kg/m3) was hung from an analytical balance with a hook for underfloor
weighing (GR–300, A & D Co., Ltd.) using a hanging wire. Sodium polytungstate solution(density: 2.71×103 kg/m3, viscosity: 15.6 mPa.s) was used as the liquid. The sodium
62polytungstate solution was Newtonian fluid. Viscosity of sodium polytungstate solution wasmeasured by the viscometer (DV–II+, Brookfield). The particle concentration was set at 10kg/m3. The laser diffraction/scattering method (MT3000EX, Nikkiso Co., Ltd.), themicroscope method (MIC–D, Olympus) and Coulter counter method were used to validate the
PSD obtained from the experiments.
For sample particle hollow glass beads K1 (Sumitomo 3M), the aluminum weighing bar(diameter: 10 mm, length: 250 mm, density: 2.70×10 3 kg/m3) was hung from an analyticalbalance with a hook for underfloor weighing (GR–300, A & D Co., Ltd.) using a hanging wire.
The laser diffraction/scattering method (Microtrac MT3000EX, Nikkiso Co., Ltd.) and themicroscope method (MIC–D, Olympus) were used to validate the PSD obtained from theexperiments. Ion exchanged water (density: 1.00×103 kg/m3, viscosity: 0.89×10–3 Pa.s) andsodium hexametaphosphate were used as the liquid phase and dispersant, respectively. The
initial volume concentrations of the hollow glass bead were set at 0.01.
For sample particle hollow glass beads S60HS (Sumitomo 3M), an aluminum weighingbar (diameter: 10 mm, density: 2.70×103 kg/m3) was hung from an analytical balance with ahook for underfloor weighing (GR–300, A & D Co., Ltd.) using a hanging wire. The weighing
bars lengths are 100 mm and 250 mm. The laser diffraction/scattering method (Microtrac
MT3000EX, Nikkiso Co., Ltd.) and the microscope method (MIC-D, Olympus) were used tovalidate the PSD obtained from the experiments. Ion exchanged water (density: 1.00×103
kg/m3, viscosity: 0.89 mPa･s) and sodium hexametaphosphate were used as the liquid anddispersant, respectively. The initial volume concentrations of the hollow glass beads were set
at 0.01.
The bottoms of the weighing bars were set from the bottom of the graduated cylinder tothe position at 10 mm. A personal computer was connected to the analytical balance, and theapparent masses of weighing bars were collected in one second interval. To avoid external
effects such as airflow and temperature changes, the experimental apparatus was placed in a
box and the temperature during the experiment was maintained at 298 K.
To prepare a suspension, 1000 ml liquid and the particles to be tested were mixed in
graduated cylinder. Using a hanging wire, which did not extend due to the weight of the bar, abar was hung from the electronic precision weighing balance. After thoroughly stirring thesuspension using an agitator, the bar was set with the balance, and this was recorded as t = 0 s.
The apparent mass of weighing bar was recorded on a personal computer. After themeasurements, we calculated the PSD of the hollow glass beads.
5.3.2 Influence of the weighing bar and vessel in the BWM on size distribution
measurement of floating particle
Figures 5.1 and 5.2 depict the experimental apparatus, which consists of a suspension
vessel, weighing bar, analytical balance, and defines the sizes of the weighing bars andvessels. The aluminum weighing bars included various sized rods (diameter: 5-35 mm, length:63

109-250 mm, density: 2.70×103 kg/m3), a square rod (12×12×210 mm, density: 2.70×10 3
kg/m3) and a sheet (1.5×25×210 mm, density: 2.70×103 kg/m3). A floating particle suspension
was placed in a graduated cylinder (50-1000 ml, diameter: 21-65 mm, SIBATA ScientificTechnology Ltd.) or a square vessel (100×100×170 mm). The weighing bars were suspended
from the analytical balance with a hook for underfloor weighing (GR-300, maximum mass:
310 g, minimum readout mass: 0.1 mg, A & D Co., Ltd.) using a hanging wire. Data werecollected in 3 s intervals using a personal computer connected to the analytical balance. Table
5.1 shows the sizes and shapes of the weighing bars and vessels. The sample particles werehollow glass beads K37 (Glass beads K37; density: 0.370×103 kg/m3; Sumitomo 3M). The
shape factor of the sample particles φ was 1, because particles were spherical. Ion
exchanged water (density: 1.00×103 kg/m3; viscosity: 0.89 mPa･s) was used as the dispersion
liquid. The concentration of the suspension was set to 3.7 kg/m3, and the temperature during
the experiments was maintained at 298 K. Additionally, the PSDs were compared to theresults from the laser diffraction/scattering method (Microtrac MT3000EX, Nikkiso Co., Ltd.)and microscopy (MIC-D, OLYMPUS Co.).
To prepare a suspension, ion exchanged water and hollow glass beads K37 were mixed
in a vessel. After thoroughly stirring with a hand-type agitator, the weighing bar wassuspended from the analytical balance via a hanging wire, and this was recorded as t = 0 s.
Collected data consisted of time and the apparent mass of the weighing bar. After themeasurements, we calculated the PSD of the sample particles based on the aforementioned
theory.
5.4 Results and Discussion
5.4.1 PSD measurement of glass beads GBL 100 (JIS Test Powders 2) and hollow glass
beads K1 and S60HS (Sumitomo 3M) by the BWM
5.4.1.1 PSD of glass beads GBL 100, JIS Test Powders 2 in heavy liquid
Figure 5.3 shows the PSDs obtained from the floating experiments using the glass
beads GBL 100 (JIS Test Powders 2) in sodium polytungstate solution. Reynolds number ofthe particles was smaller than 0.005 in the present study. Range of the glass beads was 75–125
μm. In this figure, the PSDs measured by the laser diffraction/scattering, microscope and theCoulter counter methods are indicated by the solid line, circle keys and dotted line,
respectively. The PSDs measured by the BWM were close to those measured by the laser
diffraction/scattering and microscope methods. We concluded that the PSD of the glass beads
GBL 100 (JIS Test Powders 2) could be measured in sodium polytungstate solution by theBWM.
5.4.1.2 PSD of hollow glass beads K1, Sumitomo 3M
Figure 5.4 shows the PSDs obtained from the floating experiments using the hollow
64glass beads K1. The range of particle size is 16–105 μm. In this figure, the PSDs measured by
the laser diffraction/scattering and microscope methods are indicated by the solid line andcircle keys, respectively. The PSDs measured by the BWM were close to those measured by
the laser diffraction/scattering and microscope methods.
5.4.1.3 PSD of hollow glass beads S60HS, Sumitomo 3M
Figure 5.5 shows the PSDs of the hollow glass beads S60HS measured using theweighing bar of 100 mm in length. The PSDs measured by the laser diffraction/scattering
method and the microscope method are indicated by the line and diamond key in this figure.
The PSD measured by the BWM was close to those measured by the laser
diffraction/scattering method and the microscope method. We concluded that the PSD of thehollow glass beads S60HS (Glass beads S60HS) could be measured by the BWM.
The PSD of the hollow glass beads S60HS measured using the weighing bar of 250 mm
in length is also shown in Figure 5.5. In this case, the time course change in the apparent massof the weighing bar did not become constant. The particle size calculated at 7200 s is about 12
μm, because the calculated floating velocity is 3.5×10 –3 m/s. However, the PSD using theweighing bar of 250 mm in length was close to those measured using the weighing bar of 100
mm in length.
5.4.2 Influence of the weighing bar and vessel in the BWM on size distribution
measurement of floating particle
5.4.2.1 Accuracy and repeatability of PSD measurements
As shown in a previous paper [8], hollow glass beads K37 are spherical. Feret’s
diameter as a particle size was determined using 200 particles and microscopy. Figure 5.6
shows the PSDs of hollow glass beads K37 measured by microscopy and the laser
diffraction/scattering method. Particle sizes ranged was 14-90 μm. The relative refractive
index of the hollow glass beads K37 was set to 1.51, which is same as that of soda-lime glass.
The PSDs were measured five times where each trial was conducted independently by adifferent person. Although each trial did not produce identical results, the measured PSDs
were similar. Moreover, the PSD measured by microscopy was similar to that measured by thelaser diffraction/scattering method. The data measured by the laser diffraction/scattering
method was statistically analyzed, and Table 5.2 shows the calculated values of the average,
standard deviation and variation coefficient. The range of the variation coefficient was 0.051
to 0.095. The PSD was determined using the average values.
Figure 5.6 shows that the PSDs of hollow glass beads K37 measured by the BWM weresimilar to those measured by the microscopy and the laser diffraction/scattering method. The
data measured by the BWM was statistically analyzed. Table 5.2 also shows the calculated
values of average, standard deviation and variation coefficient by the BWM. The variation
coefficient ranged between 0.016 and 0.087. The variation coefficients of the buoyancy
65

weighing-bar and laser diffraction/scattering methods were similar. Hence, the accuracy of
BWM is equivalent to that of the laser diffraction/scattering method.
Figure 5.7 shows the changes in the apparent mass with time measured using a 1000 ml
graduated cylinder (diameter: 65 mm) and a weighing bar (diameter: 10 mm; length: 250 mm).
The apparent mass of the weighing bar decreased until all the particles floated above theupper end of the weighing bar, and then the apparent mass of the weighing bar became
constant. The variation in the apparent mass was due to the change in the buoyant mass
against the weighing bar along with the particles floating.
5.4.2.2 Influence of weighing bar length
Becker et al. have reported the effects of sedimentation length [2]. In this work, thefloating length and weighing bar length are the same. Previously, we have reported that small
particles of hollow glass beads S60HS cannot be measured using a long weighing bar [9].
Figure 5.8 shows the PSDs of hollow glass beads K37 measured using the BWM by rods
with varying lengths as well as the laser diffraction/scattering method. Regardless of theweighing bar length, the PSD could be measured. The solid line denotes the PSD from theaveraged data using the laser diffraction/scattering method (Table 5.2). The red dashed andblue dashed lines denote the average particle size plus or minus one standard deviation,
respectively. The PSDs measured by the BWM are within plus or minus one standard
deviation. Hence, we conclude that the BWM can correctly determine the PSD using aweighing bar length of 109 to 250 mm.
5.4.2.3 Influence of weighing bar diameter
Figure 5.9 shows the PSDs measured using a 1000 ml graduated cylinder (diameter: 65mm) and rods with different diameters. The solid, red dashed, and blue dashed lines represent
the average particle size, average particle size plus one standard deviation, and average
particle size minus one standard deviation from the laser diffraction/scattering method,respectively. All trials using the BWM produced similar size distributions. Moreover, theresults confirm that rods with diameters between 10 and 25 mm successfully measured thePSDs of hollow glass beads K37.
Figure 5.10 shows the PSDs measured using a 250 ml graduated cylinder (diameter: 38mm) and varying diameter rods. Rods with diameters between 20 and 35 mm gave PSDs thatwere outside the standard deviation range of the laser diffraction/scattering method. Theseobservations indicate that the PSD cannot be measured using a large rod and small vessel.
5.4.2.4 Influence of vessel size
Figure 5.11 shows the PSDs measured using a rod with a 10 mm diameter and different
graduated cylinders. Regardless of the volume of the graduated cylinder, the PSDs measured
using a 210 mm long rod were almost identical. Additionally, employing a 109 mm long rod
gave the same results. The PSDs measured by the BWM are within one standard deviation of66

the average using the laser diffraction/scattering method. Thus, using graduate cylinders
between 50 and 1000 ml and 10 mm diameter rods with lengths between 109 and 210 mm canmeasure the PSD of hollow glass beads K37.
5.4.2.5 Influence of weighing bar shape
Figure 5.12 shows the PSDs measured by using the aluminum square rods and thealuminum sheet as weighing bars. The continuous line represents that distribution measured
by the laser diffraction/scattering method. The square rods and the sheet gave almost the same
PSDs when the volumes of the graduated cylinders were 500 ml and 1000 ml. In these cases,
both the aluminum square rods and aluminum sheet can be used as the weighing bar tomeasure the PSD of hollow glass beads. On the other hand, the square rod did not give thesimilar PSDs when the volume of the graduated cylinder was 100 ml. We considered that theBoycott effect influences the measurement results of PSD [5].
5.4.2.6 Influence of vessel shape and weighing bar position
To investigate the influence of vessel shape, the PSD of hollow glass beads were
measured by using the square vessel and the aluminum rod with the 10 mm diameter. The
suspension height and the rod length were 140 mm and 109 mm, respectively. The weighing
bar positions were center (50 mm-50 mm) and off-center (20 mm-20 mm) in the square vessel.
Figure 5.13 shows the PSDs using the square vessel. The continuous line indicates the
distribution measured by the laser diffraction/scattering method. The PSDs measured at thecenter were same as those measured at off-center, and agreed well with the laser
diffraction/scattering method. Hence, the location of the weighing bar does not influence thePSDs, and the square vessel can effectively measure the distribution of hollow glass beads.
5.4.2.7 Influence of the sectional area ratio
The sectional area ratio a/aC is defined by the following formulasectional area of weighing bar, asectional area ratio, a/aC .
sectional area of vessel, aC(5.9)

Figure 5.14 shows the particle sizes with 10%, 50% and 90% undersize as a function of thesectional area ratio. The solid, red dashed and blue dashed lines denote the average particle
size using the laser diffraction/scattering method, average particle size plus standard deviation,
and average particle size minus standard deviation, respectively. The error bars are thevariation coefficient of the BWM in Table 5.2. A PSD measurement is deemed feasible when
the data or the error bar is within one standard deviation of the average using the laser
diffraction/scattering method. However, the BWM is not applicable when the data and error
bars extend beyond this range.
The particle sizes increased when the sectional area ratio exceeded 0.2. This increase
has been attributed to the wall effect [10] and Boycott effect [4]. Although the wall effect has
been reported to influence the migration velocity when the size ratio of the particle to the67vessel is large, the wall effect slows the migration velocity of the particles. Because theimpact of the wall effect is the opposite of the observed trend, the influence of the wall effecton the velocity of floating particles is negligible. The Boycott effect occurs when the sectional
area ratio of pan/vessel is large. Due to a concentration difference, liquid flow accelerates the
migration velocity. Hence, the Boycott effect cannot be ignored if the sectional area ratio of
the pan and vessel exceeds 0.3 [5]. Because the migration velocity of the particles accelerates
according to the Boycott effect, the measured PSD is larger than the actual distribution.
Herein the Boycott effect cannot be ignored when the sectional area ratio of the rod and vessel
is greater than about 0.2.
On the other hand, the particle sizes of 10% undersize increased when the sectional area
ratio was less than about 0.02. In addition to the wall and Boycott effects, there may be
another factor in this case. Although this factor is currently unclear, particle size
measurements may be difficult due to the small volume of the weighing bar. Additionally, thevariation coefficient of the particle size of 10% undersize measured by the BWM is about
one-sixth of that measured the laser diffraction/scattering method. If the variation coefficientof the BWM is the same as that of the laser diffraction/scattering method, then the PSD may
be measured. Consequently, we conclude that the BWM can measure PSDs when the range ofthe sectional area ratio is 0.02-0.2.
5.5 Conclusions
Using the BWM, we measured the floating PSD and discuss the influence of particle
migration lengths as well as the sizes and shapes of the weighing bar and vessel. The
following results were determined:
1. The PSDs of hollow glass beads GBL 100 (JIS Test Powders 2) can be measured in
sodium polytungstate solution and close to those measured by the laser diffraction/
scattering and microscope methods.
2. The PSDs of the hollow glass beads K1 (Sumitomo 3M) can be measured using the BWM
and the precision of the PSD is comparable to that obtained by the laser
diffraction/scattering and microscope methods.
3. The PSDs of the hollow glass beads S60HS (Sumitomo 3M) can be measured using the
BWM and the precision of the PSD is comparable to those obtained by the laser
diffraction/scattering method and the microscope method, but when the long typeweighing bar (250 mm) was used, small particle of the hollow glass beads S60HS could
not measure in 2 h.
4. The PSDs of hollow glass beads K37 (Sumitomo 3M) can be measured by the BWM andthe result obtained is comparable to those determined by microscopy and laser
diffraction/scattering methods.
5. The influence of the weighing bar length on the PSD could not be confirmed for thehollow glass beads K37.
68

6. When the graduated cylinder was used as the vessel, the PSDs of hollow glass beads K37
could be measured by using the square rod or sheet as the weighing bar.
7. When the rod with the 10 mm diameter was used as the weighing bar, the PSDs of hollow
glass beads K37 could be measured by using the square vessel. The location of theweighing bar does not influence the PSDs.
8. The PSDs can be measured by the BWM when the sectional area ratio is 0.02-0.2.

69Nomenclature
a
aC
d
dV

sectional area of the weighing bar, m2
sectional area of the vessel, m2
rod diameter, mvessel diameter, mDcumulative mass undersize, %f(x)mass frequency of particle size x, m-1

F(ε)
g
G
GR

voidage function, gravitational acceleration, m/s2
apparent mass of the weighing bar, kg
VB ρB WR , kghhVRlength of the weighing bar, mheight of the vessel, mcumulative mass oversize, %xv ( x) ρ L
Reynolds number ( 
), –
μL

Re
t
v(x)

time, sfloating velocity of particle size x, m/s

VB
W

volume of the weighing bar, m3
buoyant mass of the weighing bar in a suspension, kg

WRW0  (W0  W ) xparticle size, m

ε
φ
μL
ρL
ρB
ρP

voidage, Wadell’s shape factor, liquid viscosity, Pa･s
liquid density, kg/m3
density of the weighing bar, kg/m3

ρS

xmax

xi

f ( x)dx , kg

particle density, kg/m3
density of the suspension, kg/m3

ζ
standard deviation, m
Subscript
i
i th
max
maximumminminimum
0
initial t = 0
10
50
90

10%
50%
90%
70

Personal
computerdV
RS-232C cable

AnalyticalbalancedHanging wirehThermal insulation
panel

hv

Graduated cylinder

Weighing bar
Heating panel
Controller

Figure 5.1 Schematic diagram of the experimental apparatus, weighing bar and vessel.

(a) rod type(b) square rod type(c) sheet typeFigure 5.2 Schematic diagrams of the weighing bars.

71Figure 5.3 PSDs of glass beads GBL 100, JIS Test Powders 2 in heavy liquid.

Figure 5.4 PSDs of hollow glass bubbles K1, Sumitomo 3M.

72

Al rod
diameter = 10 mm
v = 3.5 x 10-5 m.s-1

v = 1.5 x 10-5 m.s-1

Figure 5.5 PSDs of hollow glass bubbles S60HS, Sumitomo 3M.

Figure 5.6 PSDs measured by microscopy, laser diffraction/scattering method and BWM.

73

Figure 5.7 Time course change in the apparent mass of the weighing bar using
the 1000 ml graduated cylinder.

Figure 5.8 Influence of rod length on PSD (vessel diameter = 65 mm).

74

Figure 5.9 Influence of rod diameter on PSD (vessel diameter = 65 mm).

Figure 5.10 Influence of rod diameter on PSD (vessel diameter = 38 mm).

75

Figure 5.11 Influence of vessel size on PSD.

Figure 5.12 Influence of weighing bar shape on PSD.

76

Figure 5.13 PSDs measured in the square vessel.

100
suitable region

80
60
40
20

MT3000EX
x

90

x

x

0 -3
10

50

10

10

-2

10

-1

10

0

a/a [-]
C

Figure 5.14 Particle sizes as a function of sectional area ratio.

77

Table 5.1 Sizes of weighing bar and vessel
weighing bar

vessel (graduated cylinder)

rod diameter, d rod length, hinner diameter, D height, H (volume)

φ 5 mm

250 mm

φ 21 mm

220 mm(50 ml)φ 8 mm

250 mmφ 38 mm

305 mm(250 ml)

φ 10 mm

109 mmφ 47 mm

360 mm(500 ml)

φ 10 mm

210 mmφ 65 mm

420 mm (1000 ml)

φ 10 mm

250 mm

φ 15 mm

210 mm

φ 20 mm

210 mm

φ 25 mm

210 mm

φ 35 mm

160 mm

φ 35 mm

109 mmTable 5.2 Particle size, standard deviation and variation coefficient
MT3000EX

BWM

xav [μ m] σ [μ m] σ /xav [-]xav [μ m] σ [μ m] σ /xav [-]

10

21.02

2.00

0.095

19.73

0.31

0.016

20
30

27.71
33.59

1.84
1.95

0.066
0.058

28.02
33.28

2.44
2.71

0.087
0.082

40
50

38.54
42.67

1.97
2.62

0.051
0.061

38.06
42.15

2.81
2.22

0.074
0.053

60
70

48.23
52.41

3.04
3.71

0.063
0.071

47.20
52.56

3.04
3.83

0.064
0.073

80
90

57.41
64.96

3.74
4.86

0.065
0.075

57.82
68.89

3.50
3.54

0.061
0.051

D [%]

78

Chapter 6Size Distribution Measurement of Floating Particles in the Allen Region
by the Buoyancy Weighing-Bar Method
6.1 Introduction
Information about particle size distribution (PSD) is important to remove particles in aliquid via floatation and/or sedimentation. In this section, we aim to develop an innovative
method to measure the PSD using the buoyancy weighing-bar method (BWM). The BWM
measures the change in suspension density due to particle migration by weighing thebuoyancy against a weighing bar hung in the suspension, and subsequently calculating the
PSD using the length of the weighing bar and the time-course change in the apparent mass of
the weighing bar [1,2,6]. We have previously reported that the PSDs of floating particles canbe measured using the BWM [5]. This method has been used to measure the PSD of paraffinbeads, nylon beads, and hollow glass beads in the Stokes region. Moreover, the size of larger
particles can be measured by the sieving method or laser scattering/diffraction method.
However, the large particle sizes measured by the sieving or laser scattering/diffraction
method may not correspond to particle sizes based on the movement in a liquid. Theoretically,
the BWM should be applicable to measure the PSD in both the Stokes and Allen regions.
Although size distribution measurements of large particles in the Stokes region require special
skills to adjust the liquid properties, measurements in the Allen region do not.
Herein we experimentally investigate the applicability of the BWM to measure the PSD
of floating particles in the Allen region (2<Re<500).
6.2 Theory
In this section, the theory about floating particle refers to Chapter 1.2.2, but the particle
size x is given by the Allen formula [3]
1

3


L L
1
 225

x  vx 
2 2
φ 4  L   P  g 


(6.1)

where φ is Wadell’s shape factor, g is the gravitational acceleration, and μL is the viscosity ofthe dispersion liquid containing the dispersant. Wadell’s shape factor for a cylindrical particle
φ is calculated using Eq. (6.2) [4] asφ=surface area of a sphere with the same volume as the particle
surface area of the particle(6.2)

The floating velocity of the particles v(x) is calculated using Eq. (1.34) in Chapter 1. From
Eqs. (6.1) and (1.34), time t is an inverse function of particle size x. The PSD of thesuspended particles is calculated using the particle size at each time, and then plotting the79

corresponding undersize mass percentage. Although we employed the Allen formula insteadof the Stokes formula, this theory and the procedure are the same as those in a previous
paper [5].
6.3 Materials and Methods
The diagram of the apparatus used in the experiment is shows in Figure 2.3 in Chapter
2. Table 6.1 shows the particle and liquid properties. The sample particles were polystyrene
beads (spherical) and nylon beads (cylindrical, FNB#800, Fuji Manufacturing Co., Ltd.).
Although the shape of the nylon particles varied slightly, most particles were cylindrical witha diameter to length ratio of approximately 1.0. The particle densities of the polystyrene beads
and nylon beads were 1026 kg/m3 and 1120 kg/m3, respectively. Sodium chloride (reagent
grade, Kanto Chemical Co., Inc.) solutions prepared with ion exchanged water were used as
the liquid phase. The concentrations of sodium chloride were 10 wt% (density: 1069 kg/m3,
viscosity: 1.07 mPa･s) and 24 wt% (density: 1170 kg/m3, viscosity: 1.55 mPa･s) for
polystyrene beads and nylon beads, respectively. The initial volume concentration of thesuspension C0/ρP was set to 0.01, and the temperature during the experiment was maintained
at 298 K. Measurements lasted 150 seconds. A laser diffraction/scattering method (Microtrac
MT3000EX, Nikkiso Co., Ltd.) and a sieving method (425–2000 μm, JIS testing sieve, Tokyo
Screen Co., Ltd.) were used to validate the experimental PSD.
To prepare a suspension, 1000 ml liquid and the particles to be tested were mixed in a
graduated cylinder. The weighing bar was hung from an electronic precision weighing balanceusing a hanging wire, which did not extend due to the weight of the weighing bar. After
thoroughly stirring the suspension using an agitator, the weighing bar was set with the balance,
and this was recorded as t = 0 s. The measuring data, which consisted of time t and thecorresponding mass of the weighing bar GB, were recorded on a personal computer. After themeasurements, we calculated the PSD of the tested particles based on the above–describedtheory.
6.4. Results and Discussion
6.4.1. Spherical particles
Figure 6.1 shows the change in the apparent mass of the weighing bar GB–GB0 withtime when polystyrene particles were used. The apparent mass of the weighing bar linearlydecreased for approximately 100 s, at which point the polystyrene particles floated above theupper end of the weighing bar, and the apparent mass of the weighing bar became constantGB∞. From Eqs. (1.21) and (1.27) in Chapter 1,
V C
GB  GB0  B 0 (  P   L )

P

(6.3)
80

The volume of the weighing bar VB, initial volume concentration C0/ρP, particle density ρP andliquid density ρL were 1.96×10–5 m3, 0.01, 1026 kg/m3, 1069 kg/m3, respectively. The value ofthe apparent mass difference GB∞–GB0 calculated by Eq. (6.3) was –8.4×10–6 kg, which is
identical to the apparent mass difference of measured data. The change in the apparent mass
was due to the change in the buoyant mass against the weighing bar along with the floating
particles. Thus, we confirmed that the floating PSD of spherical particles in the Allen region
can be measured using the BWM.
Figure 6.2 shows the PSDs in the Allen region obtained from the floating experiments
using polystyrene particles as well as those measured by a laser diffraction/scattering method
and a sieving method. Theoretically, Wadell’s shape factor of the polystyrene particles was 1.0
because the particles were spherical. All three methods gave similar PSDs. The particle sizes
calculated by Eq. (6.1) were included in the Allen region. Hence, our results confirm that the
BWM can measure the PSD of floating spherical particles in the Allen region.
6.4.2. Cylindrical particles
Figure 6.3 shows the change in the apparent mass of the weighing bar GB–GB0 withtime when nylon particles were used. The apparent mass of the weighing bar linearlydecreased for approximately 30 s, at which point all the nylon particles floated above theupper end of the weighing bar, and the apparent mass of the weighing bar became constant.
The volume of the weighing bar VB, initial volume concentration C0/ρP, particle density ρP,
and liquid density ρL were 1.96×10–5 m3, 0.01, 1120 kg/m3, 1170 kg/m3, respectively. The
apparent mass difference GB∞–GB0 calculated by Eq. (6.3) was –9.8×10–6 kg. The measured
value of the apparent mass difference was –9.4×10–6 kg, which is almost the same as thecalculated value. Thus, we confirmed that the floating PSD of cylindrical particles can bemeasured using the BWM.
Figure 6.4 shows the PSDs of the nylon particles in the Allen region measured using
the BWM, laser diffraction/scattering method, and sieving method. Wadell’s shape factor of
the nylon particles was theoretically 0.873 because the ratio of the diameter to the length was
1.0. The sizes of the nylon particles calculated by Eq. (6.1) were in the Allen region. We alsomeasured the PSDs for nylon particles by a laser diffraction/scattering method and a sieving
method. Regardless of the method, the measured PSDs were similar.
6.5 Conclusions
Using the BWM, we measured the PSDs of floating solid particles in liquid phases
where the density was adjusted by the concentration of sodium chloride. The results led to thefollowing conclusions:
1. Although particles in the Allen region migrate, the floating PSDs of solid particles can bemeasured by the BWM.
81

2. The precision of the PSD by the BWM is comparable to that obtained by a laser
diffraction/ scattering method or a sieving method.

82

Nomenclature
C
D(x)
g
GB
GR
Re

solid concentration of suspension, kg/m3
undersize mass percentage of particle size x, %gravitational acceleration, m/s2
apparent mass of weighing bar, kg
VBρB–WR, kg
xv ( x) ρ L
Reynolds number ( 
), –
μL

R(x)
t
v(x)

oversize mass percentage of particle size x, %time, sfloating velocity of particle size x, m/s

VB
x
φ
μL

volume of weighing bar, m3
particle size, mWadell’s shape factor, –liquid viscosity, Pa･sρLliquid density, kg/m3
ρP
particle density, kg/m3
Subscripts
0∞

initial t = 0
infinity t = ∞83

Figure 6.1 Apparent mass of the weighing bar as
a function of time (spherical particle).

Figure 6.2 Particle size distributions of spherical particle.

84

Figure 6.3 Apparent mass of the weighing bar as
a function of time (cylindrical particle).

Figure 6.4 Particle size distributions of cylindrical particle.

85

Table 6.1 Particle and liquid properties
Particle (shape)

Particle density

LiquidDensity and viscosity

polystyrene beads (sphere)

1026 kg/m3

10% NaCl

1069 kg/m3, 1.07mPa･s

nylon beads (cylinder)

1120 kg/m3

24% NaCl

1170 kg/m3, 1.55 mPa･s

86

Chapter 7
Conclusions
The PSD of settling and floating particles can be determined by the BWM. This method
is simple in operation, low-cost of equipment and can produce the high accuracy result in
PSD measurement. The results led to the following conclusions.
7.1 Settling particles(1) The PSD of large particle up to 63 m can be measured in water using the BWM.
(2) There is the graphical analogy of PSD among Andreasen pipette, sedimentation balance,
fluidization curve and buoyancy weighing bar methods(3) BWM can measure the PSD of a needle-shaped particle for one-component or
two-component particles, and the result data obtained comparable to that measured byAndreasen pipette and settling balance methods.
(4) For measuring the PSD of the JIS Test Powder 2, No.4 (White fused alumina), thesuitable initial concentration is 10 kg/m 3 or less. The PSDs measured by the BWM wasclose to those measured by the Andreasen pipette method, but not close to thosemeasured by the laser diffraction/scattering method.
(5) BWM can measure the PSD of glass beads GBL 30 (JIS Test Powders 2) and gave theclose result to those measured by Andreasen pipette, Coulter counter and laser
diffraction/scattering methods.
(6) When a bar composed of aluminum, copper, acryl resin and SUS 304 were used tomeasure the PSD of non–spherical silica sand Class 3, JIS Test Powders 1, the materialof the weighing bar did not influence the PSD. The PSDs measured by the BWM gavethe similar result to those measured by the sedimentation balance, Andreasen pipette andlaser diffraction/scattering methods.
(7) When a bar composed of aluminum, copper and SUS 304 were used to measure the PSD
of non–spherical calcium carbonate (heavy) Class 16, JIS Test Powders 1, the material of
the weighing bar did not influence the PSD. The PSDs measured by the BWM gave thesimilar result to those measured by sedimentation balance, Andreasen pipette,microscope and laser diffraction/scattering methods.
(8) BWM can measure the PSD of magnesite (China) and soft–burned magnesia (China).
The PSD measured by the BWM agreed well to those measured by laser diffraction/
scattering method.
(9) BWM can measure the PSD of JIS Test Powders 1, class 8 (KANTO loam). The PSD
measured by the BWM agreed well to those measured by sedimentation balance,
Andreasen pipette and laser diffraction/scattering methods.
(10) BWM can measure the PSD of JIS Test Powders 1, fly ash class 5. The PSD measured
87

by the BWM agreed well to those measured by sedimentation balance and laser
diffraction/scattering methods.
(11) BWM can measure the PSD of JIS Test Powders 1, talc class 4 used the ion exchanged
water and ethanol as dispersion liquid. The PSD measured by the BWM agreed well tothose measured by sedimentation balance and Andreasen pipette methods, but not close
to those measured by laser diffraction/scattering method.
(12) The PSDs of JIS Test Powders 1, class 3, class 10 and class 11 can be measured by the
BWM within two hours, and the Rosin-Rammler distribution can be estimated.
(13) The PSD of JIS Test Powders 1, class 17 cannot be estimated because the cumulative
mass oversize is up to about 0.1 within two hours.
(14) The PSD of suspended solids in spring water obtained at a construction site can bemeasured by the BWM and its Rosin-Rammler distribution.
(15) The velocity mean size of all particles ( x0 ) and the velocity mean size of particles fromxmin to xi ( x i ) by the BWM are determined by1

 n ΔG x 2  2x0    i i 
 i 1 G 1

 i ΔGi xi2  2x i   

G
i

1




 n

 ΔGi  G 
 i 1 i ΔGi  G  Gi  i 1

where Gi is the buoyancy difference mass of particles xi.
(16) The velocity mean sizes of particles ( x i ) can be recognized graphically by the BWM.
The maximum particle size can be defined by tmax.
(17) The PSD and mean particle size of multi-component particles can be measured by the
BWM as the BWM and laser diffraction/scattering method yield similar results.
(18) The PSD and mean particle size for a ternary spherical particle mixture of glass beads(GBL 60, GBL 40 and GBL 30) can be measured by the BWM because the BWM
measured and calculated results are similar.
7.2 Floating particles(1) The PSDs of hollow glass beads K37 (Sumitomo 3M) can be measured by the BWM andthe result obtained is comparable to those determined by microscopy and laser
diffraction/scattering methods.
(2) The influence of the weighing bar length on the PSD could not be confirmed for thehollow glass beads K37.
(3) When the graduated cylinder was used as the vessel, the PSDs of hollow glass beads
K37 could be measured by using the square rod or sheet as the weighing bar.
(4) When the rod with the 10 mm diameter was used as the weighing bar, the PSDs ofhollow glass beads K37 could be measured by using the square vessel. The location of88

the weighing bar does not influence the PSDs.
(5) The PSDs can be measured by the BWM when the sectional area ratio is 0.02-0.2.
(6) The PSDs of the hollow glass beads K1 (Sumitomo 3M) can be measured using the
BWM and the precision of the PSD is comparable to that obtained by the laser
diffraction/scattering and microscope methods.
(7) The PSDs of the hollow glass beads S60HS (Sumitomo 3M) can be measured using the
BWM and the precision of the PSD is comparable to those obtained by the laser
diffraction/scattering method and the microscope method.
(8) When the long type weighing bar (250 mm) was used, small particle of the hollow glass
beads S60HS could not measure in 2 h.
(9) The PSDs of hollow glass beads GBL 100 (JIS Test Powders 2) can be measured in
sodium polytungstate solution and close to those measured by the laser diffraction/
scattering and microscope methods.
(10) Although particles in the Allen region migrate, the floating PSDs of solid particles canbe measured by the BWM.
(11) The precision of the PSD is comparable to that obtained by a laser diffraction/scattering
method or a sieving method.

89

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Acknowledgements
I would like to acknowledge and extend my heartfelt gratitude to the following persons who
have made the completion of my doctoral study:
1. My advisor Prof. Eiji Obata and co-advisor Asc. Prof Yuichi Ohira for their guidance,
understanding, patience and support during my study at Muroran Institute of Technology.
2. All my friends in Reaction Laboratory, Muroran Institute of Technology for theirunderstanding and assistance.
3. All the civitas academia of Muroran Institute of Technology for helpful and kindness
during my study at Muroran Institute of Technology.
4. All of those who supported me in any respect during the completion of the study that I
could not mention personally one by one.
5. My wife Yanti Hutapea, my daughter Ruth Stephanie Tambun and my son Raymond
Yehezkiel Tambun for their forbearance and understanding.
Finally, I would like to thank God and His son Jesus Christ, who made all things possible.

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