Size Analysis of Irregular Shaped Particles in Sieving Comparison with the Coulter Counter Howard N. Rosen' and Hugh M. Hulburt2 Department of Chemical Engineering, Il'orthwestern Unicersity,Evanston, Ill. 60201
A study was made with a Coulter counter, using potassium sulfate crystals to determine the average size of particles in nine sieve cuts ranging from 60 to 400 U. S. standard mesh. The average equivalent spherical diameter of particles in a sieve cut can b e greater than the average opening of the sieve screens. A modification of the Coulter counter sampling equipment is used for measuring particle sizes above 100 microns.
A
popular physical measurement of crystalline substances is a determination of part'icle size distribution bl- laboratory sieving, discussed by PIIulliii (1961) and Orr (1966), among others. Fundameiit'al studies of size distributions usually relate the size of particles examined to an equivalent spherical length or diameter-Le., the diameter of a spherical particle whose volume is equal to that of the particle being studied. One method used to charact'erize the average size of particles in a sieve cut is to define the average as equal to the linear average of the openings of the sieve screens-for example, a -170+200-mesh U. S.standard sieve cut with openiiigs of 88 and 74 niicrons, respectively, would be expected t'o yield a product of average equivalent length of 81 microns. If the particles were perfect spheres normal1~-distributed about 81 micron.?, this would be true. Unfortunately, in most cases the particles being analyzed are not spheres aiid usually are not normally distributed about the linear average sieve screen opening. I n this iyork, a study was made of the appropriate average screen opening to be used for estimating particle size. A relatively new particle size aiialyzer is the Coulter COUIIter, the theory and operation of which are discussed by U11rich (1960). This device is an electrical sellsing inst,rument capable of detecting particles suspended in a slightly conducting liquid as they are drawn from one chamber t o another t'hrough a small orifice. I n theory, the aniplit'ude of t'he voltage pulse produced as a particle passes the sensing device is proportional to the particle volume. This theory has been verified experimentally by Eckhoff (1967) for irregularly shaped particles. I n the present work, we studied sieve cuts of irregularly shaped particles with a Coulter counter to determine an average particle size, which was then compared to the linear average sieve opening. I n the literature, Coulter counter studies of particles larger than 100 microns have not been discussed because of the necessity to use a large 56O-micro1i-aperture tube for which the standard apparatus has inconveniently small capacity. This paper describes a technique for studying particles up to 250 microns with a 560-micron-aperture tube by using auxiliary equipment. Deflnition of Average Size
When sieve analysis is used to determine particle size distributions, it becomes necessary to define a value that charPresent address, 2033 Haste St., Berkeley, Calif. 94704.
* To whom correspondence should be sent. 658
Ind. Eng. Chem. Fundam., Vol. 9,
No. 4, 1970
acterizes the size of a given sieve cut. One such value is t'he average sieve opening between two sieve cuts, Ls. The Coulter counter determines the ciimulative oversize number distribution, N, defined as the total number of part'icles larger than size I,. h frequency dist'ribution, f, of particles in a sieve cut is defined such that fdL equals the number of part'icles having size in the range L , L dL. 1Ioments off are defined as
+
Physical parameters such as percentage oversized number,
S,size, D , surface area, A , and weight, li7, dist,ributions, can be defined in terms of the monient,s. Derived distributions directly related to physical parameters, such as oversize percentage, A', size D, surface area, A , aiid weight, TI', can be defined in terms of the frequency, f, and its moments as follows:
(3) PL
\
(4)
The distribution, A ( L ) , for example, is t,he percentage of surface area associated with particles larger than size L. Figure 1 shows typical distributions derived from an arbitrary frequency distribution plot. Values of L for which S,D , A , and TI' are 50% oversize can be defined as L,, LD,La, aud LIy, reipectively. The experimenter must determine what values of L will best describe the average size of the sample. For example, if a weight distribution is required for the over-all distribution, LIP,is used for each sieve cut. I n sieve analysis, the sieve cuts are usually narrow distributions compared t o the over-all distributions; conse-
~~
Table 1. Sieve cut, Mesh
-325 -270 -230 -200 -170 -120 -100 -80 -60
Figure 1.
+ 400 + 325 + 270
+ 230 + 200 + 170 + 120 + 100 + 80
Sieve Cut Characteristics of 3-Inch U. S. Standard Sieves Sieve
Coulter Counter Aperture Tube Sire,
Opening, Mt,O",
3744 44-53 53-63 63-74 74-88 88-125 125-149 149-177 177-250
1.7
MiW0"S
40.5 48.5 58.0 68.5 81.0 106.5 137.0 163.0 213.5
200 200 200 200 200 200 560 560 560
Physical distributions from frequency plot
quently, the values of I,,, Lu, LA,and LW are not much different from each other. In practice, the value of L, is difficult to obtain accurately from Coulter counter analysis because, at low values of L , electrical noise becomes significant. This noise is interpreted by the instrument as particle couuts and can be a significant fraction of the total count. The other distributions are weighted with a value of I, to a power of 1 or greater, and thus filter out the coutribution of noise t o the total distribution. In this study, values of LU and Lw are evaluated for several sieve cuts and compared to Ls.
Figure 2. Potassium sulfate crystals produced in laboratory vacuum crystallizer -60
+ 80-mesh cut
SLURRY MIXER
Experimental
A Model B Coulter counter with 200- and 560-micron orifices was used for the size analysis of potassium sulfate crystals obtained from a laboratory vacuum crystallizer (Rosen and Hulburt, 1969). A 6% solution of ammonium thiocyanate dissolved in 2-propanol was used as the electrolytic solution. As seen under a microscope, the potassium sulfate crystals were elongated prisms with projected areas showing a widthto-length ratio ranging from 1:7 to almost 1:l (Figure 2). The ratio varied with size, the smaller particles being more elone.nt,erl. ~~.~~ - ~ ~-~~ ~ -
Threeinch U. S. standard sieves were used with sizes of 60, 80, 100, 120, 170, 200, 230, 270, 325, and 400 mesh. A 25gram sample of crystals was screened by a mechanical vibrator for 2% hours prior to a Coulter counter analysis. Sieve cuts smaller than 120-mesh were studied by standard methods using a 200-micron-aperture tube. Those from 120- to 60-mesh were studied in a 560-micron-aperture tube with a modification to the Coulter counter sampling equipment. Figure 3 shows the auxiliary equipment used to accommodate the 560-micron-aperture tube. A special 1-gallon glass jar with a u esit line located at the bottom was used to transport the particles in the ammonium thiocyanate electrolyte solution. The particles were mixed by a stirrer using a polyethylene propeller blade a t a speed high enough to distribute the particles evenly in the solution. A tubing pump.was used to transport enough slurry to fill the 300-ml beaker used for Coulter counter analysis. With smaller aperture tubes the
Figure 3. Auxiliary Coulter counter sampling equipment for use with 560-micron-aperture tube
beaker contains enough slurry for the entire analysis; with a 560-micron tube the beaker must be refilled several times during an analysis. This auxiliary mising and transporting equipment transfers a well mised slurry quickly without damaging the particles, This procedure gives consistent results over repeated runs. Discussion of Results
Typical plots of percentage oversize size and weight distributions are shown in Figures 4 and 5. Table I lists the sieve characteristics and Table I1 gives the results of the analyses. Runs were made in triplicate and average deviation values were determiued for each cut. The ratios of LU aud L,v to the average sieve opening, K D and K F , respectively, were also determined. The 88-to 125-micron sample was analyzed with both large- and small-aperture tubes to r a k e sure t.he results were consistent. An LTVvalue of 122.6 microns was obt,ained with the 200-micron-aperture tube, compared to 125.0 in the Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
659
~~
Table II.
Coulter Counter Analysis for Average Sizes of Particles in Sieve Cuts
50% Oversize Size Averoge, Microns,
8, Microns
a
b
40.5 48.5 58.0 68.5 81.0 106.5 137.0 163.0 213.5
52.1 60.8 67.1 72.2 84.1 119.3 158.0 165.3 211.8
49.5 55.7 65.2 75.4 84.5 119.7 152.8 170.4 224.2
Figure 4.
LD, C
SIZE
~
46.4 53.8 64.9 75.1 85.7 115.2 150.0 162.3 ,
, ,
Average Value and Deviation, Microns
49.3+ 2.0 56.8+2.6 65.7100.9 7 4 . 2 i 1.4 84.8i0.6 118.11k 1 . 9 153.6 2 . 9 166.0 i 2 . 9 218.0 16 . 2
*
50% Oversize Weight
P
a
b
C
Average Value and Deviation, Microns
0,415 0.454 0.490 0.538 0.568 0.508 0.497 0,602 0.602
56.3 64.0 68.3 75.6 86.5 121.8 164.8 170.3 223.6
53.9 57.6 66.4 77.6 91.1 126.1 154.8 179.9 224.0
49.3 58.3 68.4 77.8 88.2 127.0 150.7 181.3 , . .
5 3 . 2 12.6 60.012.7 67.7*0.9 77.010.9 88.6+ 1.7 1 2 5 . 0 12.1 156.8 =t5 . 4 177.2* 4 . 6 223.8 i 0 . 2
Average, Microns, I i v
KD
1.22 1.17 1.13 1.08 1.05 1.11 1.12 1.02 1.02
Krv
1.31 1.24 1.17 1.12 1.09 1.17 1.14 1.09 1.05
0
. ri~cichi
Figure 5.
Percentage oversized size distribution
560-micron tube. Since these values are within the limits of error for the 560-micron-aperture tube, the results of the two aperture tubes appear consistent. The Coulter counter sieve cut analyses yielded values of L w or Lo with average deviation of no more than 4%. The values of Llr are always higher than Lo, as would be expected, since in a weight average the larger particles are more heavily weighted than t'he smaller ones. However, since the distributions are narrow, the observed difference between Llv and Lo is small, at most 8% for a Ls of 40.5 microns. The values of KTrand K D are larger for the smaller sieve cut's. This is entirely consistent with microscopic observation of the different sieve cuts; the elongated prisms of the smaller sizes appear to be more elongated than the larger sieve cuts. Since a particle will fit t'hrough a screen opening whose size is the same as the smallest cross section of the particle, the smaller screen sizes will appear to have over-all larger particles when converted to equivalent spheres. The ratios K D and Kr$.have been defined with respect to Ls, the arithmetic mean sieve opening of adjacent sieves. This value is often used, but if the sieve cuts are broad, the geoniet'ric mean might' be a better estimate of the mean particle size. In the present case, the cuts are narrow and the geomet'ric mean sieve opening differs only slightly from the geometric mean. Of more concern is the precision with which the size distribution can be determined by the Coulter counter. Studies reported by Edwards and Wilke (1967), Grover, Naaman, BenSasson, and Doljanski (1969), and Hurley (1970) point out a number of corrections that must be applied to t'he raw read660 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
Percentage oversize weight distribution
ings of AR, the change in resistance produced by a particle within the orifice, These arise from nonuniformity of the electric field within the einpty orifice due to end effects and nonuniform flow and from the effect of particle shape 011 the apparent conductance of the orifice containing the particle. These effects are absorbed in practice in a calibration constant determined by sizing monodisperse particles of knom-11 volume, such as the ragweed pollen used in this work. Thus, for nonconducting particles,
AR
=
sv
R, v o
The shape factor has been calculated for ellipsoids, most recently by Hurley (1970). For prolate ellipsoids, oriented with the long axis, c, parallel to the tube axis,
s
=
+
2ji2X02 - XO(XO2- 1) 111 [(X, l ) / ( X 0 - 1111 (7)
where
x, =
l/d(l - (b/c)Z)
(8)
and b and c are the semiminor and semimajor axes. For spheres, b = c, and S = 1.5. For values of b/c between 0.4 and 1.0, Equation 7 is well represented by
S
1.5[1
- 0.6(b/c)]
The particle volume for prolate spheroids is
v, = 47r3 b2c --
(9)
I n the measurements reported here, L is the diameter of a sphere having volume equal to that of the particle. Hence,
A R = -R -, a L 3 v o
Nomenclature
4
A
If the particle is in fact a prolate ellipsoid,
AR
=
=
b/c,
R2 S T (2b)3
S
I n fact, if the short axis of the sphere, 2b, is taken equal to the mesh opening, Ls, =
(2S/3~)”~
(15)
I‘sing this in Equation 9, we can estimate p from the reported values of K D ,Thus, p =
1.5,/[(3Ko3j2)
+ 0.91
average size based on 50y0weight oversize, microns percentage number oversize AR resistance change, ohms R, resistance constant, ohms S particle shape factor, dimensionless V particle volume, microns3 V, = effective orifice volume, microns3 V, = volume of prolate spheroid, microns3 p n = nth moment o f f p = prolate spheroid axes ratio, dimensionless
Lw
V o 6p
Hence,
Kn
percentage area oversize
= percentage size oversize
f = frequency distribution, no. per micron K D = LD/Ls,dimensionless Krr = Lrr./Ls, dimensionless La = average size based on 5001, area oversize, microns L D = average size based on 50% size oversize, microns L.v = average size based on 50% number oversize, microns L s = linear average sieve openings, microns
VO
=
=
D
R “SVp
Using Equation 10 and the axis ratio, p
AR
exceed the average sieve opening. The ratio of the lengths of crystal sides or axes (the crystal habit) may be a function of crystal size.
(16)
The calculated values are given in Table 11. They increase slightly with particle size and range from 0.415 for the smallest particles to 0.602 for the largest. S o quantitative significance should probably be given to this difference, although it agrees with the trend of visual observation.. Conclusions
The equirnleiit spherical sizes of particleq in a sieve cut for irregularly shaped particles (mhich realistically includes most crystalline product5) are not necessarily equal to and usually
= = = = = =
Literature Cited Eckhoff, R.K., J . Sci. Instr. 44, 648 (1967). Edwards, V. H., Wilke, C. R., Biotechnol. Bioeng. 9, 559-74 (1967’I \ - - - . ,.
Grover, X. B., Naaman, J., Ben-Sasson, S., Doljanski, F., Biophys. J . 9, 1398-414 (1969). Grover, S . B., Saaman, J., Ben-Sasson, S., Doljanski, F.,Nadav, E., Biophys. J . 9, 1415-25 (1969). Hurlev, James, Biophys. J . 10,74-9 (1970). lIulliti, J. W., “Crystallization,” Butterworths, London, England, 1961. Orr, C., “Particle Technology,” Macmillan, New York, 1966. Rosen, H. N., Hulburt, H. lI.,Waqhington meeting, American Institute of Chemical Engineers, November 1969. Ullrich, 0. A., Sew York Instrument Society of America Conference, September 1960. RECEIVED for review February 2, 1970 ACCEPTEDAugust 7, 1970
Continuous Gas Chromatography of Multicomponent Hydrocarbon Mixtures Don E. Carter’ and Gerald 1. Esterson Washington University, St. Louis, M o . 68180
A continuous gas chromatograph was used to analyze ternary and quaternary light hydrocarbon mixtures. Sample mixture is f e d continuously to the instrument. A sinusoidal variation in sample concentration i s imparted to the feed stream before i t enters the column by a piston driven in simple harmonic motion. For an n-component mixture, n - 1 measurements of phase relative to the input are made a t downstream points in the column. The composition of the sample i s calculated from these measurements. The instrument i s intended for process stream monitoring and control applications.
T h e gas chromatographs now used in process stream monitoring and control applications are essentially automated batch instruments. This batch-type instrument automatically withdraws a sample of fluid from a process stream; Prererit addres, RIonsanto Co., St. Louis, AIo. 63166. To whom corre.pondence should be sent.
injects it into a gas chromatograph; measures, records, and transmits the response of the detector to the individual components as they emerge from the column; and then starts the process over again (Fraade, 1964). I n a Belgian patent, Carter (1962) described a new, truly continuous analytical gas chromatograph. This instrument Ind. Eng. Chem. Fundam., Vol. 9, NO.4, 1970
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