Measurement of the floating particle size distribution by a buoyancy weighing-bar method

Powder Technology 201 (2010) 283–288

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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c

Measurement of the floating particle size distribution by a buoyancy weighing-bar method Takumi Motoi, Yuichi Ohira ⁎, Eiji Obata Division of Applied Sciences, Graduate School of Engineering, Muroran Institute of Technology, Muroran, Hokkaido 050-8585, Japan

a r t i c l e

i n f o

Article history: Received 17 September 2009 Received in revised form 13 April 2010 Accepted 13 April 2010 Available online 18 April 2010 Keywords: Floating particle Particle size distribution Buoyant mass Solid particle Stokes diameter

a b s t r a c t The particle size distribution of cylinder-shaped solid particles was measured using a buoyancy weighing-bar method where the liquid phase density was adjusted to settle or float the particles. The particle size distribution obtained in our experiment agrees with the particle size measured by other method. The present study demonstrates that a buoyancy weighing-bar method, a novel method for measuring the particle size distribution, is suitable for measuring the particle size distribution of a floating solid. The precision of the resulting particle size distribution is comparable to that of a laser diffraction/scattering method as well as a direct measurement with a micrometer. Moreover, this buoyancy weighing-bar method can measure the particle size distribution even in a mixture of two particles with different sizes. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Various approaches have been used to measure particle size distribution [1]. For solid–liquid systems, it is important to measure particle size distribution by the Stokes diameter. However, the Andreasen pipette method [2], sedimentation balance method [3], centrifugal sedimentation method [4], etc. can measure particle size distribution in suspensions because these methods measure the migration velocities of particles in solution and then use Stokes formula to calculate particle size. The disadvantages of these methods are that they are time consuming and require special skills. On the other hand, particle size distribution can be analyzed using a different principle through micrographs [5], laser diffraction/scattering methods [6], and coulter counter method [7]; the drawbacks to this type of method is that numerous samples must be used for accurate particle size distribution. Although laser diffraction/scattering and coulter counter methods produce highly accurate results within a shorter time, the equipment is extremely expensive. A simple and costeffective new method to determine particle size distribution is in high demand. We aim to develop a new method to measure the particle size distribution using a buoyancy weighing-bar method. In this method, the density change in a suspension due to particle migration is measured by weighing buoyancy against a weighing-bar hung in the suspension, and the particle size distribution is calculated using the length of the bar and the time course change in the mass of the bar [8].

⁎ Corresponding author. Tel.: + 81 143 46 5768; fax: + 81 143 46 5701. E-mail address: [email protected] (Y. Ohira). 0032-5910/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2010.04.015

This apparatus consists of an analytical balance with a hook for underfloor weighing, a personal computer, and a weighing-bar, which is used to detect the density change in suspension. This apparatus has a simple structure, is easy to maintain, and does not require special skills. Furthermore, it is economical, and is even capable of automeasurements. We have reported that the particle size distributions of settling particles can be measured using a buoyancy weighing-bar method (BWM) [8]. Theoretically, a BWM can be applied not only to settling particles, but also to floating particles, including bubbles and liquid droplets. However, the latter has not been verified. In this study, we experimentally investigate the applicability of a BWM on measurements of the particle size distribution of floating particle. Floating particles include microbubbles, liquid drops, and solid particles, which were smaller than liquid density. In terms of easy handling and availability of standard samples, we used solid particles in this study. 2. Theory Let us assume the particles are uniformly dispersed in suspension. Schematic diagram of particle floating is shown in Fig. 1. As shown in Fig. 1(a), the initial buoyant mass of the submerged weighing-bar WB0 depended on the particles between the top of the weighing-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: ρS0 = ρL +

C0 ðρ −ρL Þ; ρP P

ð1Þ

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GBt = VB ρB −WBt = VB ðρB −ρS Þ:

ð6Þ

As shown in Fig. 1(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 the weighing-bar WB∞ and the final apparent mass of the weighing-bar GB∞ in suspension at t = ∞ is given by the following equations: ρS∞ = ρL ;

ð7Þ

WB∞ = VB ρL ;

ð8Þ

GB∞ = VB ρB −WB∞ = VB ðρB −ρL Þ:

ð9Þ

Fig. 2 schematically illustrates the calculating method of the particle size distribution used in the present study. Eq. (10) shows the mass balance of particles in suspension [1]. Fig. 1. Schematic diagram of particle floating.

x

x

C0 −C = C0 ∫xmax f ðxÞdx + C0 ∫ximin i WB0 = VB ρS0 ;

ð2Þ

GB0 = VB ρB −WB0 = VB ðρB −ρS0 Þ;

ð3Þ

vðxÞt f ðxÞdx h

ð10Þ

From Eqs. (2), (5), (8) and (10), x

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, the volume of the weighing-bar is VB. As shown in Fig. 1(b), concentration of suspension C decreases with the time, because large particles have floated. The density of suspension ρS, the buoyant mass of the weighing-bar WBt and the apparent mass of the bar GBt in suspension at t = t is given by the following equations: ρS = ρL +

C ðρ −ρL Þ; ρP P

WBt = VB ρS ;

ð4Þ

vðxÞt f ðxÞdx h

ð11Þ

where v(x) is the floating velocity of the particle, f(x) is the mass frequency of the particle size x. Differentiate Eq. (11) with respect to the time t, we obtain: −

dW vðxÞ x = ðW0 −W∞ Þ∫ximin f ðxÞdx: dt h

ð12Þ

From Eqs. (11) and (12),


ð5Þ

x

W0 −W = ðW0 −W∞ Þ∫xmax f ðxÞdx + ðW0 −W∞ Þ∫ximin i

WBt = WRt +

 dWBt t: dt

Fig. 2. Construction method to determine the floating particle size distribution.

ð13Þ

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Table 1 Sample particles. Sample particles

Shape

Particle size [m]

Particle density [kg m− 3]

Glassbubbles, K37 Paraffin particle NB202(A) NB303(B) FNB#400(C) FNB#600(D)

Sphere Sphere Cylinder Cylinder Cylinder Cylinder

0.012–0.081 × 10− 3 0.060–0.30 × 10− 3 0.2 × 10− 3 (nominal) 0.3 × 10− 3 (nominal) 0.4 × 10− 3 (nominal) 0.6 × 10− 3 (nominal)

0.37 × 103 0.89 × 103 1.12 × 103 1.12 × 103 1.12 × 103 1.12 × 103

The apparent mass of the submerged weighing-bar GBt is given by Eq. (6). It is gradually decreases from GB0 to GB∞. The volume and the density of the submerged weighing-bar are constant value. Differentiate Eq. (6) with respect to the time t, we obtain: dGBt dWBt =− : dt dt

ð14Þ

Therefore, from Eqs. (6), (13) and (14), we obtain: GBt = VB ρB −WRt



 dGBt dGBt t = GRt + t; + dt dt

ð15Þ

where GRt = VB ρ B − WRt. Value of GRt calculates from tangent line based on Eq. (15). The cumulative mass percentage oversize is x

RðxÞ = 100∫xmax f ðxÞdx =

GB0 −GRt × 100 = 100−DðxÞ: GB0 −GB∞

ð16Þ

Particle size x is given by the following equation using Stokes formula modified the voidage function:

x=

1 ϕ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 18μ L vðxÞF ðεÞ : g ðρL −ρP Þ

ð17Þ

where, ϕ is Wadell's shape factor, g is the gravitational acceleration, μ L is the viscosity of the dispersion liquid contained the dispersant, ε is voidage, F(ε) is the voidage function. The voidage function is calculated by the Richardson and Zaki correlation [9]. −4:65

F ðεÞ = ε

;

ð18Þ

The floating velocity of the particles v(x) is calculated using Eq. (19): vðxÞ =

h ; t

ð19Þ

Fig. 4. Schematic diagram of the experimental apparatus. 1. Analytical balance. 2. Personal computer. 3. RS-232C cable. 4. Hanging wire. 5. Measuring cylinder. 6. Weighing bar, 7. Thermal insulation vessel. 8. Heating panel. 9. Controller.

where the length of the submerged weighing-bar is h and the time lapse is t. From Eqs. (17) and (19), time t is a quadratic function of the reciprocal of particle size x− 1. The particle size distribution of the suspended particles is prepared by calculating the particle size at each time, and then plotting the corresponding mass percentage undersize. This theory for the floating particles measured by a BWM is the similar to that for settling particles [8].

3. Materials and methods Table 1 describes the sample particles, which were Glassbubbles (K37, Sumitomo 3M), paraffin particle (Kanto Chemical Co., Inc.) made by spray and Fuji nylon beads (NB202, NB303, FNB#400, FNB#600, Fuji Manufacturing Co., Ltd.). The particle densities of the sample were 0.37 × 103 kg/m3 (Glassbubbles K37), 0.89 × 103 kg/m3 (Paraffin particle) and 1.12 × 103 kg/m3 (Fuji nylon beads). For example, Fig. 3 shows micrographs of sample particles. The shapes of Glassbubbles (a) and paraffin particle (b) were sphere-shaped. The shape of the nylon particles varied to a certain degree, but most of the particles were cylinder-shaped and the ratio of diameter and length is approximately 1 (c). The nylon particles were either used alone or mixed (using a volume ratio) at equivalent concentrations. Fig. 4 shows a diagram of the apparatus used in the experiment. A floating

Fig. 3. Micrographs of sample particles.

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particle suspension was placed in a 1000 ml measuring glass cylinder (diameter: 65 mm, Sanplatec Co., Ltd.), and an aluminum weighingbar (diameter: 10 mm, length: 250 mm, density: 2.70 × 103 kg/m3) was hung from an analytical balance with a hook for underfloor weighing (GR-300, minimum readout mass: 0.1 mg, A & D Co., Ltd.) using a hanging wire. A personal computer was connected to the analytical balance, and the data were collected in one second interval. To avoid external effects such as airflow and temperature changes, the experimental apparatus was placed in a box. A heater connected to the thermostat was placed within the box to maintain the desired temperature. A laser diffraction/scattering method (Microtrac MT3000EX, Nikkiso Co., Ltd.) was used to validate the particle size distribution obtained from the experiments. Table 2 describes the properties of sample liquids. Ion exchanged water (density: 1.00 × 103 kg/m3, viscosity: 0.89 × 10− 3 Pa s) was used as the liquid phase for Glassbubbles and paraffin particle. Sodium chloride (reagent grade, Kanto Chemical Co., Inc.) solutions were used as the liquid phase for Fuji nylon beads: one for the floating experiment (concentration: 24 wt.%, density: 1.17 × 10 3 kg/m 3 , viscosity: 1.55 × 10− 3 Pa s), and the other for settling experiment (concentration: 10 wt.%, density: 1.07 × 103 kg/m3, viscosity: 1.07 × 10− 3 Pa s). The liquid density was adjusted so that the density difference between the particle density and the liquid density was equal. The initial volume concentration of the suspension C0/ρP was set at 0.001– 0.01 m3-solid/m3-suspension, 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 measuring glass cylinder. Using a hanging wire, which did not extend due to the weight of the bar, a bar was hung from the electronic precision weighing balance. After thoroughly stirring the suspension using an agitator, the bar was set with the balance, and this was recorded as t = 0. The measuring data, which consisted of time t and the corresponding mass of the bar GB, were recorded on a personal computer. After the measurements, we calculated the particle size distribution of the tested particles based on the above-described theory.

Fig. 5. Apparent mass of the weighing bar as a function of time (sphere-shaped particles).

on particle size distributions of Glassbubbles K37. When the volume concentration is 0.03, the particle size distribution of Glassbubbles K37 could not be measured by the BWM. The particle size distributions measured by the laser diffraction/scattering method are indicated by the lines in Fig. 6. The particle size distributions measured by the BWM were close to those measured by the laser diffraction/scattering method. We concluded that the particle size distribution of floating sphere-shaped particles could be measured by the BWM. Brownian motion is observed for particles about 1 μm in diameter [10]. In this work, the range of the particle size is 12–600 μm. Because the particle size order is different by one or two digits, we consider that the Brownian motion effects are little.

4. Results and discussion 4.2. Cylinder-shaped particles 4.1. Sphere-shaped particles Fig. 5 shows the time course change in the apparent mass of the weighing-bar GB when Glassbubbles K37 and paraffin particle were used. The apparent mass of the weighing-bar decreased, at which point all particles floated above the upper end of the weighing-bar, and the apparent mass of the weighing-bar became constant. The change in the apparent mass was due to the change in buoyant mass against the weighing-bar along with the particle floating. Thus, we considered that the particle size distribution of floating particle can be measured using the BWM, and that the results agree with the findings in the previous report [8]. Fig. 6 shows the particle size distributions obtained from the floating experiments using Glassbubbles K37 and paraffin particle. The Wadell's shape factor of the sphere-shaped particles is theoretically 1.0. Reynolds number of the floating particles was smaller than 2 in the present study. When the volume concentration is 0.01 or less, there is no effect of volume concentration

Fig. 7 shows the time course change in apparent mass of the bar GB when nylon particle C was used. In a 10 wt.% sodium chloride solution

Table 2 Liquid properties. Liquid

Liquid density [kg m− 3]

Viscosity [Pa s]

Water 10% NaCl 24% NaCl

1.00 × 103 1.07 × 103 1.17 × 103

0.89 × 10− 3 1.07 × 10− 3 1.55 × 10− 3

Glassbubbles K37, Paraffin Particle C (settling) Particles A-D (floating)

Fig. 6. Particle size distributions of sphere-shaped particles.

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settling experiments using nylon particle C. The Wadell's shape factor of the cylinder-shaped particle ϕ was calculated using Eq. (20) [11]: ϕ=

surface area of a sphere having the same volume as the particle : surface area of the particle

ð20Þ

(upward triangle), which was used as the liquid phase for the settling experiments, the apparent mass of the weighing-bar (measured by the analytical balance) linearly increased for approximately 60 s, at which point all nylon particles settled below the lower end of the weighing-bar, and the apparent mass of the weighing-bar became constant. On the other hand, with the 24 wt.% sodium chloride solution (downward triangle), which was used as the liquid phase for the floating experiments, the apparent mass of the weighing-bar linearly decreased for approximately 90 s, at which point all nylon particles floated above the upper end of the bar, and the apparent mass of the weighing-bar became constant. Compared to the settling particles, it takes 1.5-fold longer for the apparent mass of the weighing-bar to become constant, suggesting the floating velocity is 1.5-fold slower than that of the settling velocity of the particles. Although the density difference between the particles and liquid phase is the same, this difference between the floating and settling velocities of the particles is due to the viscosity of the liquid phase, which is 1.5-fold greater in the floating experiment than that of the settling experiment. Fig. 8 shows the particle size distributions obtained from the floating experiments using all nylon particles and

The Wadell's shape factor of the cylinder-shaped particle is theoretically 0.873, because the ratio of diameter and length is 1. Reynolds number of floating particles was smaller than 2 in the present study. The comparison studies of the results of the settling/ floating experiment with nylon particle C demonstrated that the particle size distributions almost coincided. The precision of the particle size distributions of floating particle measured by the present method is comparable to that of the particle size distributions of settling particle reported previously [8]. As for nylon particles A, B, and D, they were only used in the floating experiment, but the particle size distribution was almost the same as their respective nominal sizes. Thus, we conclude the particle size distributions obtained for nylon particles A, B and D are as reliable as those for nylon particles C. We also measured the particle size distributions for nylon particles by the laser diffraction/scattering method. The particle size distributions of nylon particles measured by the laser diffraction/scattering method are indicated by the solid lines in Fig. 8. The particle size distributions measured by the BWM were similar to those measured by the laser diffraction/scattering method. Next, we used a mixture of nylon particles A and C for the floating experiment. The time course change (filled circle) in the mass of the bar is shown in Fig. 7. The apparent mass of the weighing-bar decreased over time, and then became constant. The decreasing trends for the initial 100 s, and that between 100 s and 300 s were different. As shown in Fig. 7, in the particle C (upward triangle) suspension, the decreasing trend in the apparent mass of the weighing-bar changed at 90 s, suggesting that the particle size distribution of particle C was measured in the initial 100 s, while that in the particle A suspension was measured between 100 s and 300 s. Fig. 9 shows the particle size distribution (filled circle) of the particle mixture. The nominal size of particles A and C are shown as broken lines in the figure. The inflection point was at approximately 50% of the mass percentage undersize, suggesting that both particle sizes were measured. The figure shows the particle size at 25% was approximately 0.25 mm, and that at 75% was approximately 0.48 mm. These results confirm that each particle size is successfully measured

Fig. 8. Particle size distributions of nylon particles (cylinder-shaped particles).

Fig. 9. Particle size distributions of a nylon particle mixture. Solid line: calculated line of nylon particles A and C.

Fig. 7. Apparent mass of the weighing bar as a function of time (cylinder-shaped particles).

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in the particle mixture. In addition, we calculated the particle size distribution of the particle mixture, which consisted of nylon particles A and C, from independent data obtained using nylon particle A and nylon particle C. The results are shown in Fig. 9 (solid line). There is a slight difference between the calculated results and the results obtained in the experiment, but they are close. Thus, we conclude that the particle size distribution of particles, which consists of two different sized particles, can be measured using the BWM if the particle density is the same. Taken together, the BWM is reliable for measuring a particle size distribution. 5. Conclusions Using the BWM, we measured the particle size distribution of the floating solid particles in the liquid phases where the density was adjusted with the concentration of sodium chloride. The following results were determined: (1) Using the BWM, the particle size distributions of floating solid particles can be measured, suggesting the theory established in particle size distribution of settling particle measurements can be applied to floating particles. (2) The particle size distribution for a particle mixture consisting of two different size particles can be measured using the BWM. (3) The precision of the particle size distribution is comparable to that obtained by the laser diffraction/scattering method, a representative high precision method used to measure particle size. Nomenclature C solid concentration of suspension, kg/m3 D(x) mass percentage undersize of particle size x, % f(x) mass frequency of the particle size x, − F(ε) voidage function, − g gravitational acceleration, m/s2 GBt mass of weighing tool at t = t, kg GRt VB ρ B − WRt, kg h submerged length of weighing tool, m R(x) mass percentage oversize of particle size x, % t time, s v(x) floating velocity of particle size x, m/s VB submerged volume of weighing tool, m3 WB buoyant mass of the submerged weighing tool in the suspension, kg WRt W0 −ðW0 −W∞ Þ∫xxmax f ðxÞdx, kg i

x ε ϕ μL ρL ρB ρP ρS

particle size, m voidage, − shape factor, − liquid viscosity, Pa s liquid density, kg/m3 density of weighing tool, kg/m3 particle density, kg/m3 density of suspension, kg/m3

Subscripts max maximum min minimum 0 initial t = 0 ∞ infinity t = ∞

Acknowledgements A part of this work has been supported by the Japan Science and Technology Agency (JST). The authors would like to thank our students for experimental assistance: Mr. Y. Koikeda and K. Nakano. References [1] T. Allen, Particle Size Measurement, Fourth edition, Chapman and Hall, London, 1990, pp. 345–355. [2] Society of Chemical Engineering of Japan, Chemical Engineering Handbook, 5th edition, Maruzen, Tokyo, Japan, 1988, pp. 224–231. [3] K. Fukui, H. Yoshida, M. Shiba, Y. Tokunaga, Investigation about data reduction and sedimentation distance of sedimentation balance method, J. Chem. Eng. Japan 33 (2000) 393–399. [4] M. Arakawa, G. Shimomura, A. Imamura, N. Yazawa, T. Yokoyama, N. Kaya, A new apparatus for measuring particle size distribution based on centrifugal sedimentation, Journal of the Society of Materials Science Japan 33 (1984) 1141–1145. [5] M. Kuriyama, H. Tokanai, E. Harada, Maximum stable drop size of pseudoplastic dispersed-phase in agitation dispersion, Kagaku Kogaku Ronbunshu 26 (2000) 745–748. [6] H. Minoshima, K. Matsushima, K. Shinohara, Experimental study on size distribution of granules prepared by spray drying: the case of a dispersed slurry containing binder, Kagaku Kogaku Ronbunshu 31 (2005) 102–107. [7] Y. Ohira, H. Takahashi, M. Takahashi, K. Ando, Wall heat transfer in a double-tube coal–slurry bubble column, Kagaku Kogaku Ronbunshu 30 (2004) 360–367. [8] E. Obata, Y. Ohira, M. Ohta, New measurement of particle size distribution by buoyancy weighing-bar method, Powder Technology 196 (2009) 163–168. [9] J.F. Richardson, W.N. Zaki, Sedimentation and fluidization: Part I, Trans. Inst. Chem. Engrs. 32 (1954) 35–52. [10] T. Allen, Particle Size Measurement, Fourth edition, Chapman and Hall, London, 1990, pp. 258–261. [11] T. Allen, Particle Size Measurement, Fourth edition, Chapman and Hall, London, 1990, pp. 128–140.