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Crystal Size Measurement: Comparison of the Techniques of Sieving and Coulter Counter
Powder Technology, 10 (1974) 153-156 @ Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
Crystal Size Measurement: Comparison of the Techniques of Sieving and Coulter Counter
J.W. MULLIN and H-M. ANG Department of Chemical Enginmering. University (G t. Britain)
College London,
Torrington Place, London
WC1 E 7JE
(Received February 27.1974)
SUMMARY
Carefully sieved crystals of nickel ammonium sulphate are analysed with a Coulter particle size counter. Simple correction factors are proposed to rationalize the two sizing techniques.
INTRODUCTION
During a study of the precipitation of nickel ammonium sulphate crystals from aqueous solution [ 11, a need arose to measure crystal size distributions in the range 10 pm to 1 mm. For such a wide size range it was clear that more than one method of analysis would have to be employed, and the two methods chosen were test sieving [2] and electronic counting with a Coulter Sumter 131. Sieving with standard test sieves is generally suitable for particle sizes from several millimetres down to about 40 pm, although the precision in the region below 75 pm is low, while the Coulter counter is particularly suitable for sizes in the range of 400 to 10 pm. Since there was a considerable overlap in their useful ranges, the opportunity was taken to make a comparison of these two particle-sizing techniques.
sieves; a sieve effectively measures the second largest dimension of a particle. On the other hand, a Coulter counter analysis measures the volume mean size of the particles; in any counting operation it gives the cumulative oversize number distribution, IV, i.e. the total number of particles larger than a given size, L. Defining F as the frequency function of the distribution of particles, FdL gives the number of particles in the size range L to L + dE, and moments of the distribution function are defined by Y” = j I,” FdL where p,, is the nth moment
M(L), can equations:
be obtained
the moment
L
N(L)
= l-
j FdLI jmFdL 0
(2) ,
0
First moment: L
j
LFdLI
0
using close-sieve sizes, of the aperture widths, to describe the size of between any two given
from
Zeroth moment:
MEAN PARTICLE SIZE
sieve analysis the arithmetic mean Ls, is normally used the particles retained
of the function
(n > 0). The oversize fraction on the basis of number, N(L), size, L(L), area, A(L) and mass,
L(L)=l-
In a
(1)
0
jLFdL
(3)
0
Second moment: L
A(L)
=
1 - j L2 FdL/ j L2 FdL 0
(4)
0
4
154
Third moment:
= 1 -
M(L)
j- L’ 0
FdLI j- L3 FdL
(5)
0
The median size (50% oversize_) can also be represented on number, size, area or mass bases, and any of the four median sizes L,, L, or LA, can be used to describe the L,, particles in a given distribution_ The area median size, L, , (from eqn. 4) is not a relevnnt quantity in sieve and Coulter counter size analyses. The number median size, Zjv, (rrom eqn. 2) is not used here because of inherent errors in Coulter counter analyses at small sizes (the instrument tends to pick up electrical noises which are counted as particles). This error is reduced for other distributions (the lst, 2nd and 3rd moment equations) which are weighted with a value of L” where n Z 1. As a sieve analysis essentially measures the width (the second largest dimension) of a particle, and as the first moment (eqn. 3) is weighted on a single dimension, a comparison should ideally be made between the mean sieve siz_e, L,, and the median size on a size basis, L,, as measured by the Coulter counter. However, because mass distributions (from eqn. 5) are widely used in sieving practice 121, both median sizes zL and zfil are also compared here with the mean sieve size J
L.s
.
EXPERIMENTAL
Nickel ammonium sulphate hexahydrate crystals, prepared by precipitation from nickel and ammonium sulphate solutions [I>41 I were sieved by a standard procedure [Z] to an end-point on test sieves of aperture size 420, 355, 300, 251, 210, 152, 76 and 37 pm. A Coulter counter analysis was then carried out on each sieve fraction. The operation of the Coulter counter [3,5] is based on the detection of particles suspended in a suitable conducting liquid, e.g. an aqueous solution of an electrolyte, as a sample of the suspension is drawn from one chamber into another through a small orifice which acts as the sensing zone of the instru-
ment. The orifice size selected depends on the size range to be measured. One commonly used electrolyte in Coulter counter studies is a 1% aqueous solution of sodium chloride, but it is not possible to use such a medium for water-soluble crystals, such as those of nickel ammonium sulphate being considered here. In such cases it is necessary to find an alternative inert conducting medium. It was not possible to use saturated aqueous nickel ammonium sulphate solutions since these are highly conducting. However, nickel ammonium sulphate is virtually insoluble in most alcohols and it was found that an aqueous isopropanol solution (2 parts alcohol and 8 parts water by weight) saturated with nickel ammonium sulphate at 25” C was entirely satisfactory [l] . A weighed amount (- 1 g) of nickel ammonium sulphate hexahydrate crystals was suspended in 370 ml of the electrolyte and a size distribution analysis was carried out with the Coulter counter. Each analysis was repeated to ensure that the results were compatible_ The lOOO+m orifice was used for sizing the 420- 355, 355 - 300, 300- 251, 251- 210 and 210 - 152~pm fractions. The 210 - 152~pm fraction was analysed using both the iOOO-pm and 560+m orifices as a check.
RESULTS
The percentage oversize by size and mass were calculated from eqns. (3) and (5) respectively, using an IBM 360 computer_ The crystals within a given close-sieve fraction were assumed to follow a normal (Gaussian) distribution, as indicated by the arithmetic probability plots in Fig. 1, which show (a) a mass - size typical distribution for a 210 - 152-pm sieve fraction and (b) the corresponding size - size distribution. It can be seen that > 40% of the crystals have sizes larger than the upper sieve cut size (210 pm) and about 1% have sizes smaller than the lower cut (152 pm). Similar observations were made for the other sieve fractions. A sample of the monoclinic crystals of nickel ammonium sulphate hexahydrate in the 210 - 152~pm sieve fraction is shown in Fig. 2. The volume and surface shape factors (volume = fuLs3; area = f, L, 2 ) were deter-
155
‘0°1
I..
2
. 5
10
.
.
..a*.
30
50
Pcrccntogr (Probabitiiy
.
.
9095
70
*I 9899
Oversize Scalcl
Fig. 1. Size distributions of nickel ammonium
sulphate crystals obtained from a Coulter counter analysis fora 210 - 152~pm sieve fraction;0 on a mass basis, 0 on a size basis_ The points refer to data ob-
tained on one particular run, the lines represent the
be&correlation
mined
by
for data from
microscopic
Fig. 2. Monoclinic crystals of nickel ammonium sulphate hexahydrate (210 - 152pm sieve fraction).
3 repeat runs.
measurement,
expressed variation,
giving
f, = 0.58 and f,= 4.4, i.e. the overall shape factor, F = f,lf, = 7-6.
cv=
Other characterizing shape factors measured included the elongation ratio (length/ breadth) = 1.2, flakiness ratio (breadth/depth) = 2.0 and sphericity (surface area of sphere having same volume as the crystal/surface area of crystal) = 0.77. The data in Table 1 clearly show that for particles of irregular shape (i.e. not spheres or cubes), the mean size obtained from a Coulter counter analysis does not coincide with that
in terms of the CV, as a percentage:
coefficient
100 OIL
(6)
The values of CV,,, and CV, , the coefficients of variation on mass and size bases respectively, range from about 10% to 2070, as would be expected for close-sieve fractions. Each result recorded in Table 1 is a mean of three readings, and a high reproducibility (> 95%) is obtained on both mass and size bases. Table 1 also gives values of K,, and K, , ratios of the median size on a mass basis, obtained from a Coulter counter analysis, to the average sieve aperture size (z,,, /L, ) and
from a sieve analysis_ Nor, of course, should it be expected, since the two techniques, as described above, measure two different properties of particulate matter. The spread about the median size can be
the median size on a size basis, obtained from a sieve analysis, to the average sieve aperture
TABLE 1 Size
analysis by Coultercounterofclose-sieved
Sieve
fraction (Pm)
nickel
ammonium
Average Coulter Mass basis sieve orifice size used KU (Pm) -% (Pm) (Pm)
4201355 3551300 300/251
387.5 327.5 275.5
1000 1000 1000
403.3 i: 3.1 343.3 f 2.2 290.7 2 4.4
2511210 2lOJ152 210/152 1521 76 761 37
230.5 181.0 181.0 114.0 56.5
1000 1000 560 560 560
253.3 210.0 211.0 127.6 69.1
+ 0.9 +-2.7 f 1.3 f 6.6 +z0.5
sulphate cyrstals
Size basis LL (Pm) 14.1 f 1.3 9.7 i 0.2
389.3 + 4.2 335.7 2 1.6
14.3 + 0.7 8.0 i 1.2
1.04 1.05
1.00 1.03
15.3 11.0 13.8 12.1 16.6 19.5
282.7 245.5 196.3 204.0 118.0 63.6
14.5 10.6 15.5 13.2 18.3 20.9
1.05 1.10 1.16 1.16 1.12 1.22
1.03 1.06 1.08 1.13 1.04 1.13
2 f f + f f
of
0.4 0.7 1.0 1.3 1.1 0.0
Z!I 5.1 f 1.0 2 4.9 + 2.0 f 7.3 i 0.4
+1.0 f 0.3 + 0.5 + 0.9 2 2.1 +-0.2
156
orifice for crystals in the 210 - 152~pm sieve fraction, the median size En1 only changed from 210.0 2 2.7 to 211.0 + 1.3 pm. LO-
LIST OF SYMBOLS
40. 20
-
CV
Crystal
Size.
lpm)
Fig. 3. Comparison of the size distributions of nickel ammonium sulphate crystals measured by sieves and Coulter counter. X Coulter _counter, 0 sieves (uncorrected), 0 sieves (corrected).
F K
L
E ) respectively_ K,, is always greater for any particular sieve cut, i.e. than &_ because the larger particles in the zs,, >L,, fraction are weighted heavier on a mass basis than on a size basis. For particles smaller than about 350 pm the values of KA,r and KL are > 1, consistent with the statement of Allen [ 53 that a Coulter counter analysis generally yields higher mean sizes than does a sieve analysis. The data for potassium sulphate crystals obtained by Rosen and Hulburt [67 show similar characteristics. Figure 3 shows a plot of the cumulative percentage oversize on a mass basis uersus size for nickel ammonium sulphate crystals measured by the techniques of sieving and Coulter counter. It can be seen that the two sets of data do not match in the approximate size range 100 - 310 pm, but the agreement becomes much better when the appropriate correction factors, K,,*, (from Table 1) are applied to the corresponding sieve aperture sizes. The results from the Coulter counter, on changing from one measuring orifice to another for a given analysis, are quite consistent. For example, Table 1 shows that on changing from the lOOO+m to the 560+m (EL/L,
Ls n (i
P”
of variation (standard deviation/median size) (70) frequency function of the particle distribution (eqn. 1) ratio of the median size from a Coulter counter analysis to the average sieve aperture size (Z/L,) particle size (pm) median size (corresponding to 50% oversize) (pm) arithmetic mean sieve aperture size (pm) positive integer exponent standard deviation from the median size nth moment of the distribution function coefficient
(eqn- 1) subscripts A area basis L length basis M mass basis N number basis
REFERENCES H-M. Ang, Ph. D. Thesis, Univ. of London, 1973. Test Sieving, International Standard IS0 2591, Zurich, 1973, and B.S. 17’96, British Standards Institution, London, 1974. Instruction Manual for Coulter Counter Model A (Industrial), Coulter Electronics, Ltd., Dunstable, England. J.W. Mullin and M-M. Osman, Kristall Tech., 8 (1973) 471,779. T. Allen, Particle Size Measurement, Chapman and Hall, London, 1968. H.N. Rosen and H-M. Hulburt, Ind. Eng. Chem. Fundam., 9 (1970) 658.
Crystal Size Measurement: Comparison of the Techniques of Sieving and Coulter Counter
J.W. MULLIN and H-M. ANG Department of Chemical Enginmering. University (G t. Britain)
College London,
Torrington Place, London
WC1 E 7JE
(Received February 27.1974)
SUMMARY
Carefully sieved crystals of nickel ammonium sulphate are analysed with a Coulter particle size counter. Simple correction factors are proposed to rationalize the two sizing techniques.
INTRODUCTION
During a study of the precipitation of nickel ammonium sulphate crystals from aqueous solution [ 11, a need arose to measure crystal size distributions in the range 10 pm to 1 mm. For such a wide size range it was clear that more than one method of analysis would have to be employed, and the two methods chosen were test sieving [2] and electronic counting with a Coulter Sumter 131. Sieving with standard test sieves is generally suitable for particle sizes from several millimetres down to about 40 pm, although the precision in the region below 75 pm is low, while the Coulter counter is particularly suitable for sizes in the range of 400 to 10 pm. Since there was a considerable overlap in their useful ranges, the opportunity was taken to make a comparison of these two particle-sizing techniques.
sieves; a sieve effectively measures the second largest dimension of a particle. On the other hand, a Coulter counter analysis measures the volume mean size of the particles; in any counting operation it gives the cumulative oversize number distribution, IV, i.e. the total number of particles larger than a given size, L. Defining F as the frequency function of the distribution of particles, FdL gives the number of particles in the size range L to L + dE, and moments of the distribution function are defined by Y” = j I,” FdL where p,, is the nth moment
M(L), can equations:
be obtained
the moment
L
N(L)
= l-
j FdLI jmFdL 0
(2) ,
0
First moment: L
j
LFdLI
0
using close-sieve sizes, of the aperture widths, to describe the size of between any two given
from
Zeroth moment:
MEAN PARTICLE SIZE
sieve analysis the arithmetic mean Ls, is normally used the particles retained
of the function
(n > 0). The oversize fraction on the basis of number, N(L), size, L(L), area, A(L) and mass,
L(L)=l-
In a
(1)
0
jLFdL
(3)
0
Second moment: L
A(L)
=
1 - j L2 FdL/ j L2 FdL 0
(4)
0
4
154
Third moment:
= 1 -
M(L)
j- L’ 0
FdLI j- L3 FdL
(5)
0
The median size (50% oversize_) can also be represented on number, size, area or mass bases, and any of the four median sizes L,, L, or LA, can be used to describe the L,, particles in a given distribution_ The area median size, L, , (from eqn. 4) is not a relevnnt quantity in sieve and Coulter counter size analyses. The number median size, Zjv, (rrom eqn. 2) is not used here because of inherent errors in Coulter counter analyses at small sizes (the instrument tends to pick up electrical noises which are counted as particles). This error is reduced for other distributions (the lst, 2nd and 3rd moment equations) which are weighted with a value of L” where n Z 1. As a sieve analysis essentially measures the width (the second largest dimension) of a particle, and as the first moment (eqn. 3) is weighted on a single dimension, a comparison should ideally be made between the mean sieve siz_e, L,, and the median size on a size basis, L,, as measured by the Coulter counter. However, because mass distributions (from eqn. 5) are widely used in sieving practice 121, both median sizes zL and zfil are also compared here with the mean sieve size J
L.s
.
EXPERIMENTAL
Nickel ammonium sulphate hexahydrate crystals, prepared by precipitation from nickel and ammonium sulphate solutions [I>41 I were sieved by a standard procedure [Z] to an end-point on test sieves of aperture size 420, 355, 300, 251, 210, 152, 76 and 37 pm. A Coulter counter analysis was then carried out on each sieve fraction. The operation of the Coulter counter [3,5] is based on the detection of particles suspended in a suitable conducting liquid, e.g. an aqueous solution of an electrolyte, as a sample of the suspension is drawn from one chamber into another through a small orifice which acts as the sensing zone of the instru-
ment. The orifice size selected depends on the size range to be measured. One commonly used electrolyte in Coulter counter studies is a 1% aqueous solution of sodium chloride, but it is not possible to use such a medium for water-soluble crystals, such as those of nickel ammonium sulphate being considered here. In such cases it is necessary to find an alternative inert conducting medium. It was not possible to use saturated aqueous nickel ammonium sulphate solutions since these are highly conducting. However, nickel ammonium sulphate is virtually insoluble in most alcohols and it was found that an aqueous isopropanol solution (2 parts alcohol and 8 parts water by weight) saturated with nickel ammonium sulphate at 25” C was entirely satisfactory [l] . A weighed amount (- 1 g) of nickel ammonium sulphate hexahydrate crystals was suspended in 370 ml of the electrolyte and a size distribution analysis was carried out with the Coulter counter. Each analysis was repeated to ensure that the results were compatible_ The lOOO+m orifice was used for sizing the 420- 355, 355 - 300, 300- 251, 251- 210 and 210 - 152~pm fractions. The 210 - 152~pm fraction was analysed using both the iOOO-pm and 560+m orifices as a check.
RESULTS
The percentage oversize by size and mass were calculated from eqns. (3) and (5) respectively, using an IBM 360 computer_ The crystals within a given close-sieve fraction were assumed to follow a normal (Gaussian) distribution, as indicated by the arithmetic probability plots in Fig. 1, which show (a) a mass - size typical distribution for a 210 - 152-pm sieve fraction and (b) the corresponding size - size distribution. It can be seen that > 40% of the crystals have sizes larger than the upper sieve cut size (210 pm) and about 1% have sizes smaller than the lower cut (152 pm). Similar observations were made for the other sieve fractions. A sample of the monoclinic crystals of nickel ammonium sulphate hexahydrate in the 210 - 152~pm sieve fraction is shown in Fig. 2. The volume and surface shape factors (volume = fuLs3; area = f, L, 2 ) were deter-
155
‘0°1
I..
2
. 5
10
.
.
..a*.
30
50
Pcrccntogr (Probabitiiy
.
.
9095
70
*I 9899
Oversize Scalcl
Fig. 1. Size distributions of nickel ammonium
sulphate crystals obtained from a Coulter counter analysis fora 210 - 152~pm sieve fraction;0 on a mass basis, 0 on a size basis_ The points refer to data ob-
tained on one particular run, the lines represent the
be&correlation
mined
by
for data from
microscopic
Fig. 2. Monoclinic crystals of nickel ammonium sulphate hexahydrate (210 - 152pm sieve fraction).
3 repeat runs.
measurement,
expressed variation,
giving
f, = 0.58 and f,= 4.4, i.e. the overall shape factor, F = f,lf, = 7-6.
cv=
Other characterizing shape factors measured included the elongation ratio (length/ breadth) = 1.2, flakiness ratio (breadth/depth) = 2.0 and sphericity (surface area of sphere having same volume as the crystal/surface area of crystal) = 0.77. The data in Table 1 clearly show that for particles of irregular shape (i.e. not spheres or cubes), the mean size obtained from a Coulter counter analysis does not coincide with that
in terms of the CV, as a percentage:
coefficient
100 OIL
(6)
The values of CV,,, and CV, , the coefficients of variation on mass and size bases respectively, range from about 10% to 2070, as would be expected for close-sieve fractions. Each result recorded in Table 1 is a mean of three readings, and a high reproducibility (> 95%) is obtained on both mass and size bases. Table 1 also gives values of K,, and K, , ratios of the median size on a mass basis, obtained from a Coulter counter analysis, to the average sieve aperture size (z,,, /L, ) and
from a sieve analysis_ Nor, of course, should it be expected, since the two techniques, as described above, measure two different properties of particulate matter. The spread about the median size can be
the median size on a size basis, obtained from a sieve analysis, to the average sieve aperture
TABLE 1 Size
analysis by Coultercounterofclose-sieved
Sieve
fraction (Pm)
nickel
ammonium
Average Coulter Mass basis sieve orifice size used KU (Pm) -% (Pm) (Pm)
4201355 3551300 300/251
387.5 327.5 275.5
1000 1000 1000
403.3 i: 3.1 343.3 f 2.2 290.7 2 4.4
2511210 2lOJ152 210/152 1521 76 761 37
230.5 181.0 181.0 114.0 56.5
1000 1000 560 560 560
253.3 210.0 211.0 127.6 69.1
+ 0.9 +-2.7 f 1.3 f 6.6 +z0.5
sulphate cyrstals
Size basis LL (Pm) 14.1 f 1.3 9.7 i 0.2
389.3 + 4.2 335.7 2 1.6
14.3 + 0.7 8.0 i 1.2
1.04 1.05
1.00 1.03
15.3 11.0 13.8 12.1 16.6 19.5
282.7 245.5 196.3 204.0 118.0 63.6
14.5 10.6 15.5 13.2 18.3 20.9
1.05 1.10 1.16 1.16 1.12 1.22
1.03 1.06 1.08 1.13 1.04 1.13
2 f f + f f
of
0.4 0.7 1.0 1.3 1.1 0.0
Z!I 5.1 f 1.0 2 4.9 + 2.0 f 7.3 i 0.4
+1.0 f 0.3 + 0.5 + 0.9 2 2.1 +-0.2
156
orifice for crystals in the 210 - 152~pm sieve fraction, the median size En1 only changed from 210.0 2 2.7 to 211.0 + 1.3 pm. LO-
LIST OF SYMBOLS
40. 20
-
CV
Crystal
Size.
lpm)
Fig. 3. Comparison of the size distributions of nickel ammonium sulphate crystals measured by sieves and Coulter counter. X Coulter _counter, 0 sieves (uncorrected), 0 sieves (corrected).
F K
L
E ) respectively_ K,, is always greater for any particular sieve cut, i.e. than &_ because the larger particles in the zs,, >L,, fraction are weighted heavier on a mass basis than on a size basis. For particles smaller than about 350 pm the values of KA,r and KL are > 1, consistent with the statement of Allen [ 53 that a Coulter counter analysis generally yields higher mean sizes than does a sieve analysis. The data for potassium sulphate crystals obtained by Rosen and Hulburt [67 show similar characteristics. Figure 3 shows a plot of the cumulative percentage oversize on a mass basis uersus size for nickel ammonium sulphate crystals measured by the techniques of sieving and Coulter counter. It can be seen that the two sets of data do not match in the approximate size range 100 - 310 pm, but the agreement becomes much better when the appropriate correction factors, K,,*, (from Table 1) are applied to the corresponding sieve aperture sizes. The results from the Coulter counter, on changing from one measuring orifice to another for a given analysis, are quite consistent. For example, Table 1 shows that on changing from the lOOO+m to the 560+m (EL/L,
Ls n (i
P”
of variation (standard deviation/median size) (70) frequency function of the particle distribution (eqn. 1) ratio of the median size from a Coulter counter analysis to the average sieve aperture size (Z/L,) particle size (pm) median size (corresponding to 50% oversize) (pm) arithmetic mean sieve aperture size (pm) positive integer exponent standard deviation from the median size nth moment of the distribution function coefficient
(eqn- 1) subscripts A area basis L length basis M mass basis N number basis
REFERENCES H-M. Ang, Ph. D. Thesis, Univ. of London, 1973. Test Sieving, International Standard IS0 2591, Zurich, 1973, and B.S. 17’96, British Standards Institution, London, 1974. Instruction Manual for Coulter Counter Model A (Industrial), Coulter Electronics, Ltd., Dunstable, England. J.W. Mullin and M-M. Osman, Kristall Tech., 8 (1973) 471,779. T. Allen, Particle Size Measurement, Chapman and Hall, London, 1968. H.N. Rosen and H-M. Hulburt, Ind. Eng. Chem. Fundam., 9 (1970) 658.