Gravitational sedimentation methods of particle size determination 7.1 Introduction Gravitational sedimentation methods of particle size determination are based on the settling behavior of a single sphere, under gravity, in a fluid of infinite extent. Many experiments have been carried out to determine the relationship between settling velocity and particle size and a unique relationship has been found between drag factor and Reynolds number. This relationship reduces to a simple equation, the Stokes equation, which applies at low Reynolds numbers. Thus at low Reynolds numbers the settling velocity defines an equivalent Stokes diameter which, for a homogeneous spherical particle, is its physical diameter. At low Reynolds number, flow is said to be laminar i.e. the fluid flow lines around the particle are unbroken. As the Reynolds number increases, turbulence sets in leading to increased drag on the particle so that it settles at a lower velocity than predicted by Stokes' equation. It therefore follows that, if the settling velocity of a homogeneous, spherical particle is known, its particle size can be deduced and, conversely, if the size is known the settling velocity can be determined. The drag force on a particle is orientation dependent, hence non- spherical particles settling with their largest cross-sectional area perpendicular to the flow direction will settle more slowly than similar particles settling with minimum area perpendicular to flow. It follows that an assembly of identical non-spherical particles, settling under laminar flow conditions, will have a range of settling velocities according to their orientation. Sedimentation techniques can be classified according to the principles outlined in Table 7.1. Table 7.2 lists the various procedures that have been developed according to the principle applied. Techniques in current use are
Transcript
Gravitational sedimentation methods of particle size determination
7.1 Introduction
Gravitational sedimentation methods of particle size determination are based on the settling behavior of a single sphere, under gravity, in a fluid of infinite extent. Many experiments have been carried out to determine the relationship between settling velocity and particle size and a unique relationship has been found between drag factor and Reynolds number. This relationship reduces to a simple equation, the Stokes equation, which applies at low Reynolds numbers. Thus at low Reynolds numbers the settling velocity defines an equivalent Stokes diameter which, for a homogeneous spherical particle, is its physical diameter.
At low Reynolds number, flow is said to be laminar i.e. the fluid flow lines around the particle are unbroken. As the Reynolds number increases, turbulence sets in leading to increased drag on the particle so that it settles at a lower velocity than predicted by Stokes' equation.
It therefore follows that, if the settling velocity of a homogeneous, spherical particle is known, its particle size can be deduced and, conversely, if the size is known the settling velocity can be determined.
The drag force on a particle is orientation dependent, hence non-spherical particles settling with their largest cross-sectional area perpendicular to the flow direction will settle more slowly than similar particles settling with minimum area perpendicular to flow. It follows that an assembly of identical non-spherical particles, settling under laminar flow conditions, will have a range of settling velocities according to their orientation. Sedimentation techniques can be classified according to the principles outlined in Table 7.1. Table 7.2 lists the various procedures that have been developed according to the principle applied. Techniques in current use are
360 Powder sampling and particle size determination
described here; descriptions of techniques, which are no longer used, can be found in earlier editions of this book [1]. Analytical procedures for some of these techniques are covered more fully in British Standards BS 3406 [2,3]; German Standards DIN 66111 [4] and DIN 66115 [5]; French Standards AFNOR NFX 11-681 [6] and NF 11-683 [7]; two International Draft Standards ISO/WD 13317-1 [8,9] and a Japanese Standard JIS Z8820 [10]. Two American Standards for gravitational X-ray analysis are also available, one for ceramic material in ASTM C958 [11] and one for Refactory Metals in ASTM B761 [12] (see [13]).
Table 7.1 Principles of sedimentation techniques
Suspension type Homogeneous
Line-start
Measurement principle Incremental Cumulative
Force-field Gravitational Centrifugal
In the homogeneous, incremental, gravitational technique, the solids concentration (or suspension density) is monitored at a known depth below the surface for an initially homogeneous suspension settling under gravity. The concentration will remain constant until the largest particle present in the suspension has fallen from the surface to the measurement zone (Figure 7.1). At the measurement zone the system will be in a state of dynamic equilibrium since, as particles leave the zone, similar particles will enter it from above to replace them. When the largest particle present in the suspension settles through the measurement zone, the concentration will fall since there will be no particles of this size above the zone. Thus, the concentration will be of particles smaller than the Stokes diameter and a plot of concentration against Stokes diameter is, in essence, the mass undersize distribution.
In the homogeneous, cumulative, gravitational technique, the rate at which solids settle out of suspension is determined for an initially homogeneous suspension settling under gravity (Figure 7.2). This technique is typified by the sedimentation balance, in which the balance pan can be in the suspension (Figure 7.3), or suspended in a clear liquid (Figure 7.4). With the former set-up, correction has to be made for the particles that do not fall on the pan; errors are also introduced since the particle free zone below the pan leads to convection currents. The latter technique also suffers from problems due to the motion of the pan as particles settle on it. In this system, the amount settled out consists of two parts, all particles larger than Stokes diameter and a fraction of particles
362 Powder sampling and particle size determination
smaller than this. The amount undersize is determined by carrying out an integration of the second fraction.
With the incremental, gravitational, line start technique (Figure 7.5) the suspension is floated on top of a container of clear liquid and, provided the particles fall independently, the largest particles present in the suspension will reach the measurement zone first and the measured concentration will be the concentration of this size band in the measurement zone. This technique can also be used in the cumulative mode (Figure 7.6).
In this presentation, some of the methods for sedimentation particle size analysis in current use are described. Although operating procedures are not covered here, it is stressed that two factors, more than anything else, lead to incorrect analyses. The first is incorrect sampling, since analyses are carried out on from a tenth of a gram up to a few grams and these samples must be representative of the bulk for the analyses to be meaningful. The second is dispersion: it has been said rightly that the most important factor in obtaining accurate sedimentation data is dispersion-the second most important factor is dispersion and the third is also dispersion!!
7.2 Resolution of sedimenting suspensions
The size range of particles within a detector is controlled by the height of the detector beam (A/z), hence the measurement gives the concentration between an upper and lower size limit. Assuming Stokes' law to apply, equation (6.8) may be written as:
di=k\ ' (7.1)
Differentiating with respect to h with t constant:
2dst ^dd,.\ k _dl 'St
dh t
The size resolution is given by:
^ ^ = (7.2) dst 2/?
For a hydrometer, assuming A/? = h: When 50 ^m particles are at a measurement depth of 10 cm, the bottom of the hydrometer bulb is at a
Gravitational sedimentation methods 363
depth of 15 cm and the top is at a depth of 5 cm, so that particles of size 35.4 |Lim will be entering the measurement zone and particles of size 70.7 |im will be leaving it. If the weight frequency of particles in the 35.4 to 50 |im range is balanced by the weight frequency in the 50 to 70.7 |Lim range the effect is balanced out, otherwise a bias results. The effect of such a bias is to mask peaks in multi-modal distributions. If the thickness of the measurement zone is less than one sixth of the settling depth, the size resolution of around 8% leads to tolerable errors.
link balance ' Rabatin and Gale spring balance Shimadzu balance ICI sedimentation column BCURA sedimentation column Fisher Dotts apparatus Decanting p-Back-scattering
Line-start, cumulative, gravitational sedimentation Werner and Travis method Granumeter Micromerograph MSA analyzer '
364 Powder sampling and particle size determination
7.3 Concentration changes in a suspension settling under gravity
Let a mass m^ = p v of powder be dispersed in a mass my= p^Vy of fluid, p and V being density and volume respectively.
Initially the mass concentration will be uniform and equal to:
di=k[^] (7.3)
where C(kO) is the concentration at depth h, time / = 0. Consider a small horizontal element at depth h. At the commencement of sedimentation, the particles leaving the element are balanced by the particles entering it from above. When the largest particles present in the
suspension leave the element, after settling from the surface, there are no similar particles entering to replace them. The concentration will then fall and become equal to a concentration smaller than that ofd^^ where d^^ is the size of the particle that settles at a velocity of M. The concentration of the suspension at depth h at time / may be written:
1 C(h,t) = -^-^-= \f(d)dd (7.4)
^.+v
where is the mass and v is the volume of solids in a volume Vj of fluid at a depth h from the surface of the suspension at time t from the commencement of sedimentation.
C(//,0) = — ^ — = f f{d)dd (7.5) •* •' " m m
From equations (7.3 to 7.5):
C{h,t) ^m, ^ d^^ .^g.
C(/?,0) m, V ^ ^ f f{d)Ad
Mtnin
Gravitational sedimentation methods 365
It is assumed that the difference between v and v is negligible compared with Vj-, Thus a graph of C(^,0/C'(/2,0) against d^^ gives the percentage undersize Stokes diameter by weight.
In the pipette method (Figure 7.7), concentration changes occurring within a settling suspension are followed by drawing off definite volumes, at predetermined times and known depths, by means of a pipette. The method was first described in 1922 by Robinson [14] who used a normal laboratory pipette. Various modifications were later suggested which complicated either the operating procedure or the apparatus [15]. Andreasen was the
h 1
I
l— 1
L
1
L I I —
2o| cm
m em
m cm
P CM
jO
Scale graduated in cm and mm
Bulb funnel Stopcock Suction tube Safely bulb Three way tap Outlet tube
Pipet
Sedimentation tube Constant temperature bath
10 cm
Fig. 7.7 (a) The fixed position pipette (b) the variable height pipette.
366 Powder sampling and particle size determination
first to leave the pipette in the sedimentation vessel for the duration of the analysis. The apparatus described by Andreasen and Lundberg [16] is the one in general use today.
Although, theoretically, errors can be reduced by the use of more complicated construction and operation, it is highly debatable as to whether this is worthwhile for routine analyses since conventional apparatus is reproducible to ±2% if operated with care [17].
This technique is a standard procedure since both the Stokes diameter and the mass undersize are determined from first principles. The method is versatile, since it can handle any powder that can be dispersed in a liquid, and the apparatus is inexpensive. The analysis is, however, time consuming and intensive.
7.5 Theory for the gravity photosedimentation technique
7.5.1 The Beer Lambert law
Consider a sedimentation container of width L measured in the direction of the light beam, containing the suspension of powder under analysis.
Let the incident light intensity falling on an element of thickness 8Z be / and the emergent intensity be 7-57. If the area of the light beam is A, the reduction in flux due to the presence of particles may be attributed to a fall in the overall intensity of the light beam, or a reduction in the area of the light beam (Figure 7.8). The emergent flux may be written:
(7-67)^ = {A-^A)I
^ = ^ (7.7) 7 A
where 5^ is the effective cross-sectional area of particles in the beam perpendicular to the direction of propagation. The equation holds, provided that the beam of light becomes homogeneous again between adjacent particles. Let there be n^ particles of diameter d^ in unit mass (1 kg) of powder and let the powder concentration in suspension be c (kg m"^); then, at time /, the following expression holds.
8 = -^^8^Z'^A«A' (7-8)
Gravitational sedimentation methods 367
where k^ is a shape coefficient (k^ = n/4 for spheres) J j ^ and d^^ are the diameters of the smallest and largest particles in the beam at time t and K^ is the extinction coefficient for a particle of diameter d^.
The extinction coefficient is defined as:
K = light obscured by a particle of diameter d^
light which would be obscured if the laws of geometric optics held
(7.9)
From equations (7.7) and (7.8)
h
Integrating for time / gives: (7.10)
In '' T \ x=St
= C^ Z ^x^x^xd \ t J jc=min
(7.11)
where /Q is the emergent light intensity with clear liquid in the beam and / is the emergent light intensity at time t.
l-dl ^
Fig. 7.8 For a light beam of cross-sectional areav4 intersecting a suspension the reduction in light flux, 5/ is proportional to the cross sectional area of particles in the beam 5^.
368 Powder sampling and particle size determination
The optical density {E^ of the suspension at time / is defined as:
^ J ^
E, =cZlogio e X K,k,n,dl (7.12)
Consider the small fall (A " ) in the optical density as the sedimentation time changes from / to / +A/ so that the maximum Stokes diameter in the beam changes from with an average value d^.
^^-cL\og,,{e)k,n,dl (7.13)
The cumulative distribution undersize by surface, assuming that K is constant for the restricted size range under consideration, is:
x=^St x=St ^
x=0 _ x=0
x=ma\ AEy
x=0 x=0
(7.14) It is therefore necessary to know how K varies with d in order to determine the size distribution. If this correction is not applied, the method is only valid for comparison purposes. Theoretical values of K may be used but this will also introduce errors, since the effective K values depend upon the optical geometry of the system. Calibration may also be against some external standard. The cumulative distribution undersize by weight is given by:
(7.15)
x=St
Z ^x4 jc=0
jc=max
Z "x4
\ ^ ^x^x ^ ^x
^ ^X
Gravitational sedimentation methods 369
The surface area of the powder is derivable from the initial concentration of the suspension and the maximum optical density (E^^^):
Combining with equation (7.13), bearing in mind that n^ is the number of particles of size d^ in unit weight (W= \) of powder:
x=max p
Sw= Y— (7-16) fcllog,o(e) ; ^ K^
where a^ is the surface shape coefficient, which may be assumed constant for a powder having a narrow size range.
Summating equation (7.16):
(7.17) C>^r —
kK^LXog^^ie) c
where K^ is the mean value for the extinction coefficient. For non-reentrant (convex) particles, the ratio of the surface and projected area shape coefficients {a/k) is equal to 4. For re-entrant particles, the surface obtained by making this assumption is the envelope surface area. Making this assumption equation (7.16) simplifies to:
S^^ ^'^ """^^ (7.18) K^L c
7.5,2 The extinction coefficient
The extinction coefficient varies with the optical properties of the solid and liquid that make up the suspension. Knowing these properties, it is possible to generate a relationship between the extinction coefficient and particle size using Mie or a boundary condition theory. Since unit area of 0.2 |Lim Ti02 cuts of 10 times as much light as unit area of 0.1 |im Ti02, (Figure 7.9), if no correction is applied the measured distribution for sub-micron Ti02 will be heavily weighted towards the coarser particles. The relationships hold for an infinitely small solid angle between the detector and the suspension so a correction for the geometry of the analyzer may be
370 Powder sampling and particle size determination
required. Alternatively, the instrument may be calibrated against some external standard. If no correction is made for variation in extinction coefficient {K =" \) the derived distribution is only a size dependent response and the method becomes a fingerprint method (i.e. useful for comparison purposes only).
0.2 0.4 0.6 0.8 1 Panicle size (JC) in microns
Fig. 7.9 Extinction curve for titanium dioxide in water for white light.
7.5.3 Turbidity measurements (Turbidimetry)
Turbidimetry measurements, using monochromatic light, yields data that can be used to determine particle size distributions. It requires simple optical technology, but complex computational software to handle the Mie theory conversion. The sensitive diameter range for latex-water suspensions was found to be 0.1 to 10 |Lim. Different types of sensors have been conceived and applied to various experimental situations. The method is particularly useful in crystallization experiments. Other applications include agglomeration, attrition and nucleation studies. Applications of the equipment and software to studies of emulsions, fumes and aerosols are also envisaged [18].
7.5.4 The photosedimentation technique
The photosedimentometer combines gravitational settling with photoelectric measurement. The principle of the technique is that a narrow
Gravitational sedimentation methods 371
horizontal beam of parallel light is projected through the suspension at a known depth on to a photocell. Assuming an initially homogeneous suspension, the attenuation at any time will be related to the undersize concentration.
ac»6.0
Fig. 7.10 Polar light scattering diagrams [19]. The outer curve magnifies the inner by a factor of 10 in order to show fine detail, x = (nD/A) where D = particle diameter and A is the wavelength of light.
Superficially, the attenuation is related to the random projected areas of the particles. The relationship is more complex than this however, due to the breakdown in the laws of geometric optics so that complex diffraction, scattering, interference and absorption effects have to be considered. For small particles, an amount of light flux, equal in magnitude to that incident upon the particle, is diffracted away from the forward direction (Figure 7.10), making their effective obscuration area twice their projected
3 72 Powder sampling and particle size determination
area. As the particle size increases, the diffracted light is contained in a decreasing solid angle in the forward direction. No matter how small the light detector, most of the diffracted light is accepted and the effective obscuration area becomes the same as the projected area. For partially transparent particles, some of the incident light is absorbed and some refracted to cause constructive and destructive interference in the transmitted beam.
It cannot therefore be assumed that each particle obstructs the light with its geometric cross-sectional area. These effects are compensated for by inclusion of an extinction coefficient {K) in the equation, making the apparent area K times the geometric area.
Early experimenters [20,21] were either unaware of, or neglected, this correction. Some research workers used monochromatic light and determined K theoretically [19,22] others used empirical calibration by comparison with some other particle sizing technique. Rose and Lloyd [23] attempted to define a universal calibration curve. Allen [24,25] designed a wide angle scanning photosedimentometer (WASP) which accepted forward scattered light so that K was constant down to a size of around 3 fim. Weichert [26] determined a relative extinction coefficient by the use of different wavelengths and speeded up the analysis by the use of different settling heights
7.5.5 Commercial photosedimentometers
Kemsis K200 is a white light, narrow angle, scanning photosedimentometer. Caron et. al [27] described an application of this instrument for studies of turbidity and sedimentation measurements of solid-liquid dispersions.
Sedimage 1000 uses a white light source and a linear sensor with 2048 detectors to determine the concentration gradient within a settling suspension. A linear image sensor, 28.6 mm in length, measures the transmitted light along this distance. The length of the image sensor limits the measurable height of particle sedimentation. The instrument continuously monitors the changing concentration of the settling suspension and the analysis is deemed complete when the smallest particle present passes the upper detector. After about 5 minutes the particle size distribution of a carborundum powder having a median size of 5 |Lim can be accurately determined [28] whereas conventional scanning photosedimentometers take around 20 minutes to carry out an analysis
A method of determining the particle size distribution from a single measurement, with a digital image acquisition system, on a sedimenting
Gravitational sedimentation methods 3 73
suspension has been presented [87]. Individual particles illuminated by a laser light sheet are tracked by a continuously operating CCD camera. The projected areas, shape factors and the centers of gravity are detected during the sedimentation process from a series of images with a constant time spread. As the algorithm is based on single particle tracking, the heterogeneity of the sample can be taken into account. From these measured particle characteristics particle sizes and settling rates are determined.
-6 Reflected ?• \ /' Light
\ -Cuvet
Fig. 7.11 The Retsch Paar Lumosed, depths of beams are marked in mm.
Retsch PAAR Lumosed (Figure 7.11), operates in the gravitational size range, with three light sources at different depths to speed up the analysis [29,30].
A range of cuvette photocentrifuges which also operate in the gravitational mode are also available commercially. With these instruments a K factor, obtained either theoretically or experimentally, can be inserted in the software algorithm.
A photosedimentometer has also been described for measuring free-falling diameters up to 20 cm in size with an application using coke. The system has been used to accurately measure a range of materials [31].
7.5.6 Sedimentation image analysis
The basic idea of this method is the analysis of particles settling under gravity using a digital image acquisition system. A continuously operating CCD camera, with a frame grabber, tracks individual particles illuminated
374 Powder sampling and particle size determination
by a laser light sheet. The laser light sheet is arranged vertically at about one-third from the bottom of the sedimentation vessel. Since the particles in the beam need to be representative of the whole sample, the analysis is over when the largest particles settle below the measurement volume. Consequently, short measurement times are not only desirable but necessary. The projected areas, shape factors, circularities and centers of gravity are determined, together with the settling velocities, during the sedimentation process from a series of images with constant time spread. The capacity of this method was demonstrated first for nearly spherical particles [32]. However it was found that problems arise when particles deviate strongly from spherical shape [33]. Later developments overcame these problems and allowed the determination of particle shape [34]. For every settling particle the projected area diameter is obtained from several images at different positions of the particle and a mean value calculated. Consequently the influence of particle shape on the sizing technique is greatly reduced. The number of detected particles is in the range of 3000 to give a reliable (number) distribution. The sizing technique was verified with BCR materials and applied to the particle size analysis of soils.
7.5.7 Transmission fluctuation spectrometry
If an extinction measurement is made with high spatial and temporal resolution, the transmitted signal shows significant fluctuations. The strength of fluctuation is related to the physical properties of the suspension and the process of spatial and temporal averaging. This connection has been exploited to calculate the particle size distribution and particle concentration from transmission measurements. A theory of temporal transmission fluctuations has been developed for the case where the beam diameter was much smaller than the particle diameter [35]. This was expanded to provide an analytical solution to the use of a focused laser beam where the beam diameter is different for each monolayer according to the variation of the beam cross-section along the path length [36].
7.6 Theory for concentration determination with the x-ray gravitational sedimentation technique
A natural extension to the use of visible radiation is to use x-rays. In this case the x-ray density is proportional to the weight of powder in the beam. The Beer-Lambert law takes the form:
I = IQQXP(-BC) (7.19)
Gravitational sedimentation methods 3 75
where J5 is a constant related to the atomic number of the powder in suspension, c is the powder concentration, / is the emergent flux with suspension in the beam and /Q is the emergent flux with clear suspension liquid. E, the x-ray density, is defined as:
E = -\og{III^) (7.20)
For powders of low atomic number, c needs to be high in order to obtain a large enough signal. Thus for silica powders, atomic number 13, a volume concentration of around 3% may be necessary and this can lead to hindered settling.
7.6.1 X-ray sedimentation
Brown and Skrebowski [37] first suggested the use of x-rays for particle size analysis and this resulted in the ICI x-ray sedimentometer [38,39]. In this instrument, a system is used in which the difference in intensity of an x-ray beam that has passed through the suspension in one half of a twin sedimentation tank, and the intensity of a reference beam which has passed through an equal thickness of clear liquid in the other half, produces an in-balance in the current produced in a differential ionization chamber. This eliminates errors due to the instability of the total output of the source, but assumes a good stability in the beam direction. Since this is not the case, the instrument suffers from zero drift that affects the results. The 18 keV radiation is produced by a water-cooled x-ray tube and monitored by the ionization chamber. This chamber measures the difference in x-ray intensity in the form of an electric current that is amplified and displayed on a pen recorder. The intensity is taken as directly proportional to the powder concentration in the beam. The sedimentation curve is converted to a cumulative percentage frequency using this proportionality and Stokes equation.
The introduction of more stable x-ray sources and detectors resulted in the development of simpler, commercially viable systems. Kalshoven [40,41] described an x-ray instrument which used a special program for scanning the sedimentation tank. As the concentration measurements by means of x-ray attenuation are very rapid, the scanning greatly speeds up the analysis, reducing the measurement time down to a few minutes. In this instrument it was done in such a way that the concentration, and hence the cumulative mass percentage undersize, is recorded as a function of Stokes diameter rather than time. An x-ray tube was used as a source and
376 Powder sampling and particle size determination
a scintillation counter as a detector. The difference in intensity between a measurement beam and a reference beam, in which the emitted beam is alternately split, was measured by a rotating wedge that automatically set the difference to zero. Sub-micron particles can be measured if the sedimentation tank is spun in a centrifuge for some time. The time integral of the centrifugal force is measured and the tank is scanned after the centrifugation. The inventor claimed that volume concentrations in the range 0.01% to 1% could be used, depending on the atomic number of the analyzed material. Experimentally it was found that readings could be taken at short distances below the surface without seriously affecting the results. When the centrifuge was used the results were independent of the time of centrifugation but no comparison analyses were presented.
Several commercial instruments utilizing these principles were developed. Oliver et. al. [42] patented a gravitational x-ray particle size analyzer that incorporated the absorption technique and improved the system described by Kalshoven. This instrument was described by Hendrix and Orr [43] and is available commercially as the Micromeretics' Sedigraph 5000 (Figure 7.12). The instrument automatically presents results as a cumulative mass percentage distribution , and the sedimentation tank is driven in such a way that the concentration is recorded directly as a function of Stokes diameter. An air cooled, low power x-ray tube is used for generation of x-rays. These are collimated into a narrow beam that passes through approximately 0.14 in, (3.6 mm) thickness of suspension. The sedimentation tank is only 1.375 in (35 mm) high, is closed at the top and, in use, completely filled with suspension. Filling and emptying of the tank is accomplished with a built-in circulating pump. The transmitted radiation is detected as pulses by a scintillation detector, these are amplified and discriminated to eliminate low energy extraneous noise. The pulses are next clipped to constant amplitude and fed to a diode pump circuit which, in conjunction with an operational amplifier with a diode feedback, gives a voltage proportional to the logarithm of the x-ray intensity and is therefore proportional to the powder concentration. The instrument can analyze powders with atomic numbers greater than 13, but rather high initial volume concentrations of powder have to be used for powders having low x-ray adsorption (0.5% to 3%). This is due for the need for the initial decrease in x-ray intensity to be greater than 20% of the intensity with clean liquid in order to obtain reasonable resolution of the resulting attenuation curve. An absolute system is used here, in which the initial intensity is first measured with clean liquid in the cell and the zero set; the suspension is then introduced. This assumes an excellent stability of the source that, in the case of x-ray tubes, may be unreliable. The authors
Gravitational sedimentation methods 377
claim a good reproducibility and present several comparison analyses with microscopy etc. The range is claimed to be 300 to 0.1 |Lim for most powders. This lower limit is unreal since it is generally accepted that gravitational sedimentation is limited to particle coarser than around a micron due to the effects of Brownian motion. The acceptance by the manufacturers of this lower size has affected the instrument design in that only a 35 mm fall-height is possible and this restricts the upper size limit [44].
Outlet
X-iay tube
SUt movement
Detector
Inlet
Digital position translator
Relative concentration signal
Pump
Sample or
pure liquid
CeU positioning signal
1 ^ 50 5 0.5 Particle size in microns
Digital program computer
Digital-to-position translator
Fig. 7.12 The Micromeretics Sedigraph.
Sedigraph 5100 was later designed with three scanning speeds, slow, standard and fast. Micromeretics' Windows-compatible operating software permits automatic overlaying of plots, saves and recalls sets of run conditions, and can operate as many as four Sedigraphs from a single computer. The temperature of the suspension is controlled automatically with heaters located in the mixing chamber. The mixing chamber is situated outside the instrument, thus eliminating the need to switch off the x-rays when changing samples, which applies to the earlier version.
The Sedigraph 5100 is also available with the MasterTech 051 Autosampler so that as many as eighteen samples can be analyzed unattended.
378 Powder sampling and particle size determination
Quantachrome Microscan reduced the time for an analysis from about 45 to about 25 minutes. In this instrument the source and detector scanned up the sedimentation tank rather than the other way round; this was claimed to reduce vibration.
In 1970 Allen and Svarovsky [45-47] developed an instrument in which the traditional x-ray tube was replaced by an isotope source. Allen and Svarovsky's design was incorporated in the Ladal x-ray scanning gravitational sedimentometer and the x-ray centrifugal sedimentometer, which are no longer commercially available. A later design of Allen's is available as the Brookhaven BI-XDC. This can operate in both the gravitational and centrifugal mode which greatly increases the size range covered. In the centrifugal mode the size range of nominal 0.2 |im titanium dioxide was found to be much narrower than in the gravitational mode and a 'phantom' bimodality appeared in the gravitational analysis which was ascribed to thermal diffusion i.e. Brownian motion.
Many industries have large data banks on product size distributions by sieve analysis and want to continue using this form of presentation. In order to accommodate this need Cho et. al [48] converted Sedigraph data to sieve data using wet screened powder in the 38 to 53 \xm size range and fitted the data to a logarithmic distribution to give the slope and median size. This procedure must be use with caution since the conversion factors are shape dependent and a new calibration is required for each product.
A review of these and other methods of size analysis is contained in a thesis by Svarovsky [49], papers by Svarovsky and Allen [50], and by Allen and Davies [51].
7.7 Relationship between density gradient and concentration
Following from equation 7.6: Let ^/z,/) be the density of the suspension at depth h and at time /. Then:
<l>{h,t)--
(/>(h,t) = Ps^'s+Pf^f
">"/
Gravitational sedimentation methods 379
^^h,t)=^''-^^'^'^-'^>'^ V,+Vf
(t>{h,t) = pf+-Ps ^s+^f
Ps
Also
Ps
Therefore
CXM.^.^Ml^ (7.21) C(/z,o) " (l>{KO)-Pf
where ^ /s the mass fraction undersize d^^.
A plot of 100 xN^(/?,/)-py)/(^(/2,0)-/7y^M against d^^ gives a
cumulative mass percentage undersize curve.
7.8 Hydrometers and divers
7.8.1 Introduction
The changes in density of a settling suspension may be followed with a hydrometer, a method widely used in soil science and in the ceramic industry. A suspension of known concentration is made up and the hydrometer inserted. Some operators leave the hydrometer in the suspension throughout the analysis and some remove it after each reading and replace it slowly before the next. Objections can be raised to either procedure since, in the former, particles settle on the hydrometer causing it to sink to a lower level than it would otherwise sink whereas, in the latter,
380 Powder sampling and particle size determination
the suspension is disturbed after each reading. To minimize errors some operators re-shake the container after each reading.
7.8.2 Theory
With the hydrometer immersed, its weight W equals the weight of suspension displaced. Let the length of stem immersed in clear suspending liquid be Z, i.e. the same as would be immersed at infinite time; the length immersed at the commencement of the analysis be L^^ and the length immersed at time t be L^. Then, at the commencement of the analysis:
W = V(f>{h,Q) + L^ap^ (7.22)
At time / = oo (clear liquid in the container)
W = Vpj+Lapf (7.23)
During the analysis, at time t
W = V(t){h,j) + L,ap2 (7.24)
where Fis the volume of the hydrometer bulb, a the cross-sectional area of the stem and hj the depth of the hydrometer bulk at time t.
Since the density of the suspension around the stem (pj, pj^ p^ varies negligibly compared with the variation in L equation (7.21) can be written:
L,-L ^ (/>{h,,t)-py
LQ-L (/>{h^,0)-pf
</> = ^^^LZJ^ (7.25) WQ-W
where w is the specific gravity marked on the hydrometer stem. If the suspension is made up of W gram of powder making up 1 L of
suspension, equation (7.25) can be written:
Gravitational sedimentation methods 381
Fig. 7.13 Depth of immersion using a hydrometer.
1000 A ( A - / ^ / ) ^
W Ps-Pf (7.26)
where p^ is the density of the suspension at time /. An equivalent formula fpr powders that are present as slurries in water removes the necessity for drying out the slurry. A specific gravity bottle is filled with water and weighed; the water is replaced with the slurry under test and the bottle is re-weighed; the difference in weights being /SW. The sample is then taken out of the bottle and used for the analysis. The equivalent formula is:
mQ(p,-pf)
^w
7.8,3 Depth of immersion
(7.27)
With the hydrometer technique, both density and depth of immersion vary with each reading. If the temperature is maintained constant at the hydrometer calibration temperature the density may be read directly from the hydrometer stem otherwise a correction needs to be applied [52].
382 Powder sampling and particle size determination
150-170
did
Fig. 7.14 Hydrometer (Calibration in gml"^ at20°C. All dimensions in mm).
It is clear that a hydrometer with a long bulb does not measure density at a point; it only measures the average density of the suspension displaced by the hydrometer. The difficulty lies in determining the point of reference below the surface to which this density refers, for when the hydrometer is placed in the suspension the liquid level rises in the container, thus giving a false reference point (Fig. 7.13).
Gravitational sedimentation methods 383
If the cross-sectional area of the container is A, the depth to be used in Stokes equation, from geometrical considerations, is [53]:
L = L^+-\ L2--\ (7.28) 2 V Aj
Several workers, who claim that corrections have to be applied for the density gradient about the bulb, and the displacement of suspension by the stem, have challenged this simple formula. Johnson [54] for example, gives the sedimentation depth, in cm, as:
^ 2 -0.5 (7.29)
7.8.4 Experimental procedure
The changes in density of a sedimenting suspension may be followed with a hydrometer (Figure 7.14), a method still used in the ceramic industry. In order to achieve sufficient accuracy in the specific gravity readings, it is necessary to use a concentration of at least 40 g L'^, which is well into the hindered settling region. The only justification for this that has been advanced is that the method gives reproducible results.
Most hydrometers are calibrated to be read at the bottom of the meniscus and this is usually not possible when the hydrometer is immersed in a suspension. The readings are, therefore, taken at the top of the meniscus and an experimentally determined correction, which is usually of the order of 0.003 g ml"^ applied.
It is usual to disregard calibration errors although these may be substantial. Good quality hydrometers are usually guaranteed to ±0.0005 g ml"^ which corresponds to an error of around ±1.5% under normal operating conditions. Johnson [54] recognized this error and suggested that it should be determined at several points by calibration in a series of dilute suspensions of common salt. A correction to meniscus reading error and density should also be applied if a wetting agent is used. The resolution, see Section 7.2, is particularly poor for the hydrometer method of size analysis where the height of the measurement zone is of the same magnitude as the depth of immersion in the suspension. Although the hydrometer cannot be recommended as an absolute instrument, it is useful for control work with wide size range continuous distributions. It is of little use with discontinuous size distributions since these give sharp
384 Powder sampling and particle size determination
boundaries in the settling suspension, which lead to peculiar results. The method for carrying out a hydrometer analysis is given in BS 1377 [55].
7.8.5 Divers
Divers overcome many of the objections associated with the hydrometer technique. These miniature hydrometers were developed by Berg [56] for use with both gravitational and centrifugal sedimentation, but have never been widely used. Basically, divers are small objects of known density that are immersed in the suspension so that they find their density level.
Berg's divers for example, were hollow glass containers that contained mercury to give the desired density. The density was then adjusted to the desired value by etching with hydrofluoric acid. Various modified divers were later developed, the final ones, by Kaye and James [57], being metal coated polythene spheres which were located with search coils.
The principle of this method is the determination of the rate at which particles settle out of a homogeneous suspension. This may be done by extracting the sediment and weighing it; allowing the sediment to fall on to a balance pan or determining the weight of powder still in suspension by using a manometer or pressure transducer.
One problem associated with this technique is that the sediment consists both of oversize (greater than Stokes diameter) and undersize particles so that the sedimentation curve of amount settled (P) against time (0 has to be differentiated to yield the weight {W) larger than Stokes diameter. Several balance systems, based on this equation, have been described.
7.9.2 Theory
The theory given below was developed by Oden [58] and modified by Coutts and Crowthers [59] and Bostock [60]. Consider a distribution of the form:
W= \ f{d)dd d=dst
Gravitational sedimentation methods 385
where W is the mass percentage having a diameter greater than Stokes diameter. The weight percentage P, which has settled out in time t, is made up of two parts: One consists of all the particles with a falling speed equal or greater than u^^, the other consists of particles with a smaller falling speed which have settled out because they started off at some intermediate position in the fluid column (Figure 7.2). If the falling velocity of one of these particles is u, the fraction of particles of this size that will have fallen out at time t is ut/h, where h is the height of the suspension. Hence:
"max '^V
P= ^f{d)dd+ ljfid}dd (7.30)
Differentiating with respect to time and multiplying by /:
'^'Ijm^ (7.31,
I.e.
dP P = W-\-t— (7.32)
dt
Since P and / are known, it is possible to determine fusing this equation. It is preferable, however to use the equation in the following form [61].
dP P^W + -^^^ (7.33)
din/
Several methods of applying this equation have been suggested. The most obvious is to tabulate / and P and hence derive dP, dt and finally W. Alternatively, P may be plotted against t and tangents drawn. A tangent drawn at point {P^^t^ will intercept the abscissa at W^, the weight percentage oversize dr. Another method is to tabulate P against t at times such that the ratio of {t/dt) remains constant, i.e. at time intervals in a geometric progression; a simple expression relating W and P then develops [62].
386 Powder sampling and particle size determination
Weighing mechanism
mmttrntrnM RMiMniiRRlMii L-J
Adjustable balance clamp
\
I
\::::1
Thermal jacket
Pressure equalizing tube
Sedimentatioii column
Balance pan
(a) (b)
Fig. 7.15 (a) Sedimentation balance with pan in the suspension, (b) Sedimentation balance with pan in clear liquid (Leschonski modification of the Sartorius balance)
Many powders have a wide size distribution and, in such cases, the time axis becomes cramped at the lower end or unduly extended at the upper end; in such cases equation (7.33) should be applied. Evaluation proceeds from a plot of P against In t; tangents are drawn every half-unit of In t\ the point where the tangent cuts the ordinate line one In t unit less than the value at which it is tangential gives the weight percentage oversize W at that value [63].
7.9.3 Sedimentation balances
In the Gallenkamp balance [64,65] the pan is placed below a sedimentation chamber with an open bottom and the whole assembly is placed in a second chamber filled with sedimentation liquid so that all the powder falls on to the pan. The weight settled is determined from the deflection of a torsion wire, and either, the run continues until all the powder has settled out of suspension, or a second experiment is carried out to determine the supernatant fraction. Problems arise during the charging operation with leakage into the clear water reservoir and particle adhesion to the premixing tube.
Gravitational sedimentation methods 387
In the Sartorius balance [66-68] the pan is suspended in the suspending liquid and a correction has to be applied for the particles which fall between the rim of the pan and the sedimentation vessel. In this instrument, when 2 mg of sediment has deposited, electronic circuitry activates a step-by-step motor which twists a torsion wire to bring the beam back to its original position. A pen records each step on a chart. The manufacturers suggest that about 8% of the powder does not settle on the pan. Leschonski [69] and Leschonski and Alex [70] reported losses of between 10% and 35%, depending on the fineness of the powder; the difference was attributed to the pumping action of the pan as it re-balances.
Leschonski modified the instrument (Figure 7.15) by placing the pan at the bottom of a sedimenting column surrounded by a second column of clear liquid so that all the powder settled on to the pan. This eliminated powder losses and resulted in more accurate analyses [71].
The manufacturers of the Cahn micro-balance make available an accessory to convert it into a sedimentation balance [72]. The balance pan is immediately below the sedimentation cylinder in order to eliminate convection currents. Shimadzu also make a beam balance [73] that operates using a simple compensating system that is prone to considerable error.
Yodshida et.al [74] describe an improved sedimentation balance. They compared the results using this balance with those from microscope counting, using three kinds of standard reference beads, and found good agreement. Fukui et al investigated data reduction and sedimentation distance for sedimentation balances [75}.
7.9.4 Sedimentation columns
Sedimentation columns (ICI, BCURA) have also been described in which the sediment is extracted, dried and weighed. A full description of these and other sedimentation columns may be found in [1]
The Horiba cuvette photo(centri)fuge has been operated in this mode [76] but is not recommended since it is very difficult to make up a stable two-layer system in a cuvette.
388 Powder sampling and particle size determination
If the powder is initially concentrated in a thin layer floating on the top of a suspending fluid, the size distribution may be determined by plotting the fractional weight settled against the free falling diameter.
7.11.2 Methods
Marshall [77] was the first to use this principle. Eadie and Payne [78] developed the Micromerograph, the only method in which the suspending fluid is air. Brezina [79,80] developed a similar water based system, the Granumeter, which operated in the sieve size range, and was intended as a replacement for sieve analyses.
The Werner and Travis methods [81,82] also operate on the layer principle but their methods have found little favor due to the basic instability of the system; a dense liquid on top of a less dense liquid being responsible for a phenomenon known as streaming in which the suspension settles en masse in the form of pockets of particles which fall rapidly through the clear liquid leaving a tail of particles behind.
Whitby [83] eliminated this fault by using a clear liquid with a density greater than that of the suspension. He also extended the size range covered by using centrifugal settling for the finer fraction. The apparatus enjoyed wide commercial success as the (Mines Safety Appliances) MSA Particle Size Analyzer although it is less widely used today [84]. The MSA analyzer can be operated in the gravitational mode, although it is more usually used in the centrifugal mode. Several papers have been published on applications of this equipment.
The line-start technique has also been used to fractionate UO3 particles by measuring the radioactivity at the bottom of a tube, the settled powder being washed out at regular intervals without disturbing the sediment [85].
2 British Standard 3406, Determination of particle size distribution, Part 2 Gravitational methods, 360
3 British Standard 3406, Determination of particle size distribution, Part 6, Centrifugal methods, 360
4 DIN 66111, Particle Size Analysis, 360
Gravitational sedimentation methods 389
5 DIN 66115 Particle Size Analysis, 360 6 NFX 11-681 Test Methods for Particle Size Analysis-Particle Size Analysis by
Gravity Sedimentation in a Liquid Medium, 360 7 NF 11-683 Test Methods for Particle Size-Analysis-Particle Size by Variable
Height Gravity Sedimentation in a Liquid-Method Using X-ray Adsorption Measurements, 360
8 ISO/WD 13317-1 Determination of Particle Size Distribution by Gravitational Liquid Sedimentation Methods- Part 1 (1996) General Principles and Guidelines, 360
9 ISO/WD 13317-3 Determination of Particle Size Distribution by Gravitational Liquid Sedimentation Methods- Part 3 (1996) The X-ray Gravitational Technique, 360
10 JIS Z8820 General Rules for the Determination of Particle Size distribution by Sedimentation in Liquid, 360
11 ASTM C958 (1997), Standard Test Method for Particle Size-Analysis of Alumina or Quartz by X-ray Monitoring of Gravity Sedimentation, 360
12 ASTM B761 (1998), Test Method for Particle Size-Analysis of Refractory Metals and their Compounds by X-ray Monitoring of Gravity Sedimentation, 360
13 Jillavenkatesa, A, Dapkunas, S.J. and Lum, L-S, H. (2001), Particle Size Characterization, NIST Sp. Publ. 960-1, 360
14 Robinson, G. W. (1922), J. Agr. ScL, (12)3, 305-321, 365 15 Andreasen, A.H.M. (1928), Kolloid Beith, 27, 405, 365 16 Andreasen, A. H.M. and Lundberg, J. J. V. (1953), Ber Dt. Keram. Ges., 11(5),
312-323,5^5 17 Allen, T. (1969), Powder Technology, 2(3), 132-140, 365 18 Crawley, G., Coumil, M. and Benedetto, D.D. (1997), Powder Technol 91(3),
197-208,577 19 Vouk, V. (1948), Ph D thesis, London University, 372, 373 20 Jarrett, B.A. and Heywood, H. (1954), Br. J. Appl Phys., Suppl. No 3, S21, 373 21 Morgan, V.T. (1954), Symp. Powder Metall, Iron and Steel Inst., Preprint
Group 1,33-38, 575 22 Lewis P.C. and Lothian, G.F. (1954), Br. J. Appl. Phys., Suppl. No 3, S571,
373 23 Rose, H.E. and Lloyd, H.B. (1946), J. Soc. Chem. Ind, 65, 52, 373 24 Allen, T. (1968), Powder Technol., 2, 132-140,, 373 25 Allen, T. (1968), Powder Technol., 2, 141-153, 373 26 Weichert, R. (1981), Proc. Particle Size Analysis Conference, 303-310, ed.
Allen, T. and Stanley-Wood, N.G. publ.John Wiley, 373 27 Caron, P., Fancompre, B., Membrey, F. and Foissy, A. (1996), Powder
Technol., H9,9\-\00, 373 28 Ma, X. and Wan, M. (1997), Part. Part. Syst. Charact., 14, 267-271, 373 29 Staudinger, G., Hangl, M. and Peschtl, P. (1986), Proa Partec Nurnberg, publ.
NiimbergMesse, 374
390 Powder sampling and particle size determination
30 Staudinger, G., Hangl, M. and Peschtl, P. (1986), Part. Part. Syst. Charact., 3, 158-162,57^
31 Marquard, H. von (1998), Aufbereit Tech., 39(9), 461-466, Eng., 374 32 Hubner, T., Will, S. and Leipertz, A. (1998), Proc. 7'^ European Symp. Particle
Charact., Partec '98, Numberg, 243-253, 375 33 Hubner, T., Will, S. and Leipertz, A. (1999), Part. Part. Syst. Charact., 16, 85-
91,575 34 Hubner, T., Will, S. and Leipertz, A. (2001), Part. Part. Syst. Charact., 18, 70-
78, 375 35 Breitenstein, M, Krauter, U. and Riebel, U. (1999), Part. Part. Syst. Charact.,
16, 249-256,, 375 36 Breitenstein, M, Riebel, U. and Shen, J. (2001), Part. Part. Syst. Charact., 18,
134-141,575 37 Brown, J.F. and Skrebowski, J.N. (1954), Br. J. Appl. Phys., Suppl No 3, S27,
376 38 Conlin, S.G. et. al. (1967), J. Scient. Instrum., 44, 606-610, 376 39 Nonhebel, G. (1964), ed. Gas Purification Processes, Newnes, 376 40 Kalshoven, J. (1965), British Patent, BP 1 158 338, 376 41 Kalshoven, J.(1967), Conf. Proc. Particle Size Analysis, Soc. Analyt. Chem.,
London, 376 42 Oliver, J.P., Hicken, G.K. and Orr, C. (1969), US Patent 3,449,567, 377 43 Hendrix, W.P. and Orr, C. (1970), Proc. Conf. Particle Size Analysis, Soc.
44 Borothy, J. (1975), Chimia, 29(5), 240-242, 378 45 Allen, T. (1970), Br. Patent, 1764/70 3, 379 46 Allen, T. and Svarovsky, L. (1970), J. Phys E, 3, 458-460, 379 47 Allen, T. and Svarovsky, L. (1970), Proc. Particle Size Analysis Conf.
Bradford, Publ. Soc. Anal. Chem., 379 48 Cho, H., Yildirim, K. and Austin, L.G. (1998), Powder TechnoL, 95(2), 109-
117, 57P 49 Svarovsky, L. (1972), PhD thesis, Univ. Bradford, Yorkshire, UK., 379 50 Svarovsky, L. and Allen, T. (1973), Paper presented at Heywood Memorial
Symp., Univ. Loughborough, UK., 379 51 Allen, T. and Davies, R. (1989), 4th European Symp. Particle Charact.,
110,5^2 53 Edwald, P. (1942), Ind Engg Chem., Analyt. edn., 14, 66, 383 54 Johnson, R. (1956), Trans. Ceram. Soc, 55, 237, 384 55 British Standard 1377, The Hydrometer Method of Particle Size Measurement,
384 56 Berg, S. (1940), Ingen Vidensk., Skr. B., No 2, Phys. Suppl. No 3, S27, 385 57 Kaye, B.H. and James, G.W. (1962), Br. J. Appl. Phys., 13, 415, 385 58 Oden, S. (1916), KolloidZ., 18, 33-47, 385
Gravitational sedimentation methods 391
59 Courts, J. and Crowthers, E. M. (1925), Trans. Faraday Soc, 21, 374, 385 60 Bostock, W. (1952), JScient. Instrum., 29, 209, 385 61 Gaudin, A. M., Schumann, R. and Schlechter, A. W. (1942), 1 Phys, Chem.,
46, 903, 386 62 Kim, S. C , Schlotzer, G. and Palik, E. S. (1967), Powder Technol, 1, 54-55,
386 63 Bostock, W. (1952), JScient. Instrum., 29, 209, 386 64 Bostock, W. (1952), JScient. Instrum., 29, 209, 387 65 Cohen, L. (1952), Instrum. Pract., 13, 1036, 387 66 Gerstenberg, H. (1957), Chem. Eng. Tech., 8, 589, 387 67 Bachman, D. (1959), Dechema Monograph, 31, 23-51, 387 68 Gerstenberg, H. (1959), Dechema Monograph, 31, 52-60, 387 69 Leschonski, K. (1962), Staub, 22, 475-486, 388 70 Leschonski, K. and Alex, W. (1970), Proc. Int. Symp. Particle Size, Bradford,
236-254, publ. Soc Anal. Chem. ed Groves and Wyatt-Sargent, 388 71 Pretorius, S.T. and Mandersloot, W.G.B. (1967), Powder Technol., 1, 23-27,
388 72 Kaye, B.H. and Davies, R. (1970), Proc. Conf. Particle Size Analysis,
Bradford, 207-222, publ. Soc Anal. Chem. ed Groves, M. W. G. and Wyatt-Sargent, J., 388
73 Suito, E. and Arakawa, M. (1950), Bull. J Chem. Res.,Kyota University, 23, 7, 388
74 Yoshida, H., Masuda, H., Futui, K, and Tokunaga, Y. (2001), Adv, Powder Technol., 12(1), 79-94, 388
75 Fukui, K., Yoshida, H., Shiba, M. and Tokunaga, Y. (2000), J Chem. Japan, 33(3), 393-399, 388
76 Hofftnan, R.L. (1991), J Colloid Interf. ScL, 143, 232, 388 11 Marshall, C.E. (1930), Proc. Royal Soc, A126, 427, 389 78 Eadie, F.A. and Payne, R.E. (1954), Iron Age, 174, 99, 389 79 Brezina, J.J. (1969), Sediment. Petrol, 16, 27-31, 389 80 Brezina, J. (1970), Proc. Conf. Particle Size Analysis Bradford, publ. Society
Anal. Chem. ed. Groves, M.W.G. and Wyatt-Sargent, J.L, 389 81 Werner, D. (1925), Trans. Farad Soc, 21, 381, i^P 82 Travis, P.M. (\940),ASTMBull., 29, 102, 389 83 Whitby, K.T. (1955), Heat. Pip. Air Cond, Jan., Part 1, 231, June, Part 2, 139,
