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Relationship between particle size distributions by number and weight
Short
Relationship between particle size dibotions nmnber and weight
commlmications
by
During investigations into the brittle fracture of spheres. particle size distributions by number and by weight_ together with particle shape Factors have been accurately determined_ These measurements have enabled the relationship between number and weight distribution to be put to a rigorous test. The following definitions are required_ W is the cumulative weight of particles smaller than size x, and the total sample weight is H& The cumulative fraciional undersize distribution by weight is W;llV,= Y(x). Particle counts must concern the number of particles larger than size x; this is N(x). If the mean volume of a particle in the size range x to s + dx is L’,the volume shape factor f(_x) is given by a=x3f(x). Some heterogeneous materials-oal. for example-have been found to display a change in density with particle size’, and to provide a general treatment of the problem G(X) is the density of particles in the size range x to x i- d-x. From the abore definitions d W/dx = -f(x)
G(X)X~ [dN(x)/dx]
(1)
or 1% = -f(x)e(x)x
3 dN(x)/dx dY(x),dx
(2)
The negative sign is required because N(x) decreases as x increases. The product f(x)o(x) is the weight shape factor which will be designated by g(x); this value is more convenient to determine than its separate components, especially when relatively small quantities of material are available_ For homogeneous materials, G(X) is constant so that f(x) is proportional to g(x). In general, the following relationships have been found to apply under a wide range of conditions covering the nature of the sample under test and the manner of its breakage Y(x) = (xrlk)” N(x)=ux-“-c g(x)=px_S Powder Technology - ELsevierSequoia SA,
(3) (4) (5)
Equation (3) is the Gates’-Gaudin3-SchuhmannJ equation in which m and k are the distribution and size moduli respectively_ a, c, k, m, n, p and s are parameters determined from experimental data All are positive. Substituting the last three equations into eqn. (2) gives rv, =
(unpF~/m)x3-‘“~“-s’
(6)
Since all terms in this equation but x are constant it follows that m+n+s=3 and ll& = anpP/m
(7) (8)
It can be shown by integration that the weight between sizes xr and x2 (x2 > x1) is given by
w;_, = 3_yys) [x~--)-x:--q
(9)
which reduces to eqn. (8) when xr = 0, and xt = k Five l-in. diameter glass spheres were broken under slow compression taking care to avoid secondary breakage (single fracture conditions’). The lunar shaped fragments bearing the external spherical surface (exoclastic material’) were mainly contained in the debris larger than 3-mesh Tyler6 (0.263 in.). This material, accounting for approximately three quarters of the weight of the five broken spheres, was removed and sieve size analysis was performed on the material smaller than 3-mesh (iV,=25.827 g)_ Tyler standard6 mesh S-in. diam. screens were employed with hand screening in preference to mechanical procedures6; hand control can be less severe than machine methods, thus reducing particle breakage on the screen while ensuring high screening efficiency The size fractions were weighed, and the number of particles in each fraction counted, so that the average weight of a particle in the several size ranges could be calculated for shape factor determinations The entire population of particles between the screen sizes 3 and 48-mesh (0.263-0.0116 in.) were counted, a total of 10,978 particles weighing 24.728 g ; that is 95.7 % of the sample subjected to size analysis. It is not convenient to count particles below 48-mesh (0.0116 in.) manually ; other methods are available for deal-
Lausam e - Printed in the Netherlands
59
SHORT COMMYIJX~CATIOXS
ing with finer si~es~.~. The representative size assumed for particle shape factor determinations was the geometric mean between adjacent screen sizes. Extraordinary care was exercised during all stages of the experimental procedure to avoid losses and particle breakage. The computed data in its final form are shown in Fig. 1; the equations were fitted by least squares and the constants are as follows: a=6.151, c=i4, k=O.236 in, m=1.050, n=1.684, p=11.632, s= O-269_ Theoretical predictions for the endoclastic part of the fragments resulting from single fracture claim that rn= 1.0; in the present case the sample consisted of approximately 55 o/o endoclastic, the remainder being exoclastic material_ Equation (7) is satisfied to the second decimal place, since m+n+s=3.003. W& calculated from eqn. (9) for x1 =0_0116 in. and x,=0236 in. is 24.294 g, while the experimental value was 24.728 g. The difference of 0.434 g can be accounted for by the fact that the
1,
0.02
0.03
NOMINAL
0.05 SCREEN
O-1
a2
0.3
INCHES
SIZE. I
Fig- 1. Size and shape factor distributions of fragmmu below 3 mesh for five 1 in. diam. glass spheres crushed by slow compression.
3-mesh point- is slightly displaced towards the coarser region, that is to the right of k (Fig_ 1); it is a common finding that eqn. (3) begins to lose validity in the extreme coarse sizes_ On the other hand. 3c2=0.185 in. (4-mesh) fits exactly onto the line, and IV,_, calculated for x1 =0.0116 in. to xz=0.1S5 in. is 18.584 ,e which differs from the experimental value of 18.545 g by only 0.2°k. W, calculated from eqn. (8) is 25.194 g and from eqn. (9) is 25.377 g_ while the weight of the entire sample was 25.827 g. The approximately half gram difference between the measured and computed values of W, is accounted for partly in the coarse region as just discussed_ and partly in the fine region. The calculated value of tVo is an extrapolation based on the measured range only; differences from the true values are due to deviations from eqns. (3), (4) and (5) outside of the measured sizes. In the present work the shapes of the particles were prismoidal and sphenoidal. Thus. the negative exponent in eqn. (5) indicates that particle shape changes from flaky to more regular forms as the size diminishes_ This was confirmed by microscope measurements. It was also found that particles become more elongated with decreasing particle size; this is a further reason for the negative exponent in eqn. (5). Several investigators have reported that shape factor is a function of particle size’-*, and there is other evidence in the literature supporting the view that shape factor does not remain constant as particle size changesq*lo_ On the other hand, theoretical derivations of size distribution equations based on statistical-mechanical considerations frcquently assume that particleshape remains wnstani with char&g particle size”-‘* ; transformations from number to weight distributions are thus performed with a wnstant shape factor- The self consistent nature of the data presented here supports the view that particle shape can change with size in the relatively warse size region. There is some evidence that changes in the particle shape factor diminish as particle size becomes smaller. Particle shape factors are not measured routinely in wmminution studies, an omission which if remedied could yield useful information_ It is emphasized that the present findings refer only to the relatively coarse size region. Methods are available for particle shape factor determination in the fine sieve range’ (500-50 m), and a method for counting particles in the region from 50 to 1 p can be adapted to provide other data necessary for shape Powder
Technot
2 (1!%53;69) B-60
60
SHORT
COMMUNI
factor determination. Finally, it is stressed that the degree of accuracy achieved in satisfying eqns. (7), (8) and (9) is principally due to the favonrable form of the shape factor distribution curve (Fig. 1) ; other materials and other modes of breakage do not give such little deviation from a straight line. C. C. HARFUS
AXGZ
G.
STAMBOLTZIS
Henry Krunzb School of Mines, Colwnbk University, New York ( U.S.A.)
2 3 4 5 6 7 8 9 10 11 I? I3 14 15
REFERENCES 1 C. C. HARRIS .ahD H. G. S~IH. Second Spnp. Preporafion. Unirersiry of Leeds, 1957. Paper 9.
CATIONS
OR Coal
The importance of temperature control in Conlter conuter dysis Materials described as insoluble usually have a finite solubility. and in particle size analysis, where very dilute suspensions are required it is often necessary to dilute with a saturated solution of the suspended solid. When applying this procedure in this laboratory to ball-milled phenacetin, considerable variations occurred between successive readings on the Coulter Counter, at the same size setting It was observed that a small rise in temperature, brought about by the aperture illumiuatiug lamp, coincided with a fall in the count_ The effect of temperature was therefore investigated further, and extended to other materials. Experimental IlZateriaIs. Phenacetin and acetylsalicylic acid were of B-P. quality and isopropanol of general purpose reagent quality. Samples of griseofulvin and procaine penicillin were obtained from Glaxo La-. boratories Ltd., phenothiazine in aqueous suspension from 1.C.L Ltd, and “Goulac- from Production Chemicals (Rochdale) Ltd. Procedure_ Phenacetin, acetylsalicylic acid, griseoftrlvin and procaine penicillin macrocrystals were size -graded using BS. sieves. A closely graded fraction of each was wet ball-milled in the manner described previouslyl. Nonidet P42 and Dispersal T were added to the mill as wetting and dispersing agents for pheuacetin, Tween 80 for acetylsalicyIic acid and Goulac for griseofirlvin. Procaine penicillin was milled in acetone.
A. 0. GAITS, Trims. AIhfE, 51 (1915) 875. A M_ GAUDW, Trims_ AIhfE, 73 (1926) 253. R ScxnnmAh%v. AIME Tech. PubI. No. 1189.1940. J. J. GILVARRY AhD B. H. BEFK-0x1, 1. A&. PJq-s., 32 (1961) 400. R D. CADPartide Si=e Deferminmion. Interscience. N.r, York, 1955. C. C. HARRIS, Nature, 187 (1960) 401. L. W. NEEDHAS%ASD N. HILL, Fuel. 14 (1935) 222. G. G. BROW, Unit Operarions, Wiley. New York, 1950. U. N. BHRASX AXD J. H_ BROWN. Tams AIhfE. 223 (1962) 248. J. G. BES~XTT. 1. Znx Fuel. 10 (1936) 22 A. B. MAXXISG. J_ INI. FzteI_, 25 (1952) 31_ J. J. GILVARRY. J. Appl. PIyr., 32 (I 961) 39 I_ A. M.GAI_DXSA~DT_ P. MELOY, Trmrr. AIME,t23(1%2)40. R. R. KLDIPEL fihD L. G. AUSITK. Truns. AIhZ&, 232 (1965) 88.
Received December 29, 1967
Determination of solubility. Preliminary experiments showed that solutions of phenacetin in 50% ethanol gave an absorption maximum at 248 rnp which obeyed Beer‘s Law. An excess of phenacetin was heated in water and cooled to tbe required temperatitre- A sample of the clear solution was withdrawn, accurately diluted with 50% ethanol and assayed by measuring its extinction at the maximum. Griseofulvin and phenothiazine were examined in a similar manner using the peaks at 2945 mp and 253 mp respectively. Phenothiazine solubility results were very scattered, probably becanse it was very difficult to wet and therefore required an excessively long time for complete saturation- The solubility was of the order of 1 x 10m3 per cent at WC and 27J°C. Saturated solutions of acetylsalicylic acid in isopropanol were prepared in the same way, but were assayed by evaporating to dryness in a vacuum oven at 80°C and weighing the residue. Procaine penicillin was determined by the iodometric procedure of the British Pharmacopoeia. Aqueous solubilities of acetylsalicylic acid at various temperatures have already been determined2. Results are summarized in Table l_ Particle size determination. A Coulter Counter model A was used with 30-p and 70-p orifice tubes and O-OS-ml and 0_5-ml manometer volumes respectively- The orifice tubes were calibrated in the normal way using monosized pollen grains. Background counts in all the experiments were small The temperature was controlled during analysis by a 150-ml jacketed vessel. Purified water was cirPowder Technol_ 2 (1968/69)
58-60
Relationship between particle size dibotions nmnber and weight
commlmications
by
During investigations into the brittle fracture of spheres. particle size distributions by number and by weight_ together with particle shape Factors have been accurately determined_ These measurements have enabled the relationship between number and weight distribution to be put to a rigorous test. The following definitions are required_ W is the cumulative weight of particles smaller than size x, and the total sample weight is H& The cumulative fraciional undersize distribution by weight is W;llV,= Y(x). Particle counts must concern the number of particles larger than size x; this is N(x). If the mean volume of a particle in the size range x to s + dx is L’,the volume shape factor f(_x) is given by a=x3f(x). Some heterogeneous materials-oal. for example-have been found to display a change in density with particle size’, and to provide a general treatment of the problem G(X) is the density of particles in the size range x to x i- d-x. From the abore definitions d W/dx = -f(x)
G(X)X~ [dN(x)/dx]
(1)
or 1% = -f(x)e(x)x
3 dN(x)/dx dY(x),dx
(2)
The negative sign is required because N(x) decreases as x increases. The product f(x)o(x) is the weight shape factor which will be designated by g(x); this value is more convenient to determine than its separate components, especially when relatively small quantities of material are available_ For homogeneous materials, G(X) is constant so that f(x) is proportional to g(x). In general, the following relationships have been found to apply under a wide range of conditions covering the nature of the sample under test and the manner of its breakage Y(x) = (xrlk)” N(x)=ux-“-c g(x)=px_S Powder Technology - ELsevierSequoia SA,
(3) (4) (5)
Equation (3) is the Gates’-Gaudin3-SchuhmannJ equation in which m and k are the distribution and size moduli respectively_ a, c, k, m, n, p and s are parameters determined from experimental data All are positive. Substituting the last three equations into eqn. (2) gives rv, =
(unpF~/m)x3-‘“~“-s’
(6)
Since all terms in this equation but x are constant it follows that m+n+s=3 and ll& = anpP/m
(7) (8)
It can be shown by integration that the weight between sizes xr and x2 (x2 > x1) is given by
w;_, = 3_yys) [x~--)-x:--q
(9)
which reduces to eqn. (8) when xr = 0, and xt = k Five l-in. diameter glass spheres were broken under slow compression taking care to avoid secondary breakage (single fracture conditions’). The lunar shaped fragments bearing the external spherical surface (exoclastic material’) were mainly contained in the debris larger than 3-mesh Tyler6 (0.263 in.). This material, accounting for approximately three quarters of the weight of the five broken spheres, was removed and sieve size analysis was performed on the material smaller than 3-mesh (iV,=25.827 g)_ Tyler standard6 mesh S-in. diam. screens were employed with hand screening in preference to mechanical procedures6; hand control can be less severe than machine methods, thus reducing particle breakage on the screen while ensuring high screening efficiency The size fractions were weighed, and the number of particles in each fraction counted, so that the average weight of a particle in the several size ranges could be calculated for shape factor determinations The entire population of particles between the screen sizes 3 and 48-mesh (0.263-0.0116 in.) were counted, a total of 10,978 particles weighing 24.728 g ; that is 95.7 % of the sample subjected to size analysis. It is not convenient to count particles below 48-mesh (0.0116 in.) manually ; other methods are available for deal-
Lausam e - Printed in the Netherlands
59
SHORT COMMYIJX~CATIOXS
ing with finer si~es~.~. The representative size assumed for particle shape factor determinations was the geometric mean between adjacent screen sizes. Extraordinary care was exercised during all stages of the experimental procedure to avoid losses and particle breakage. The computed data in its final form are shown in Fig. 1; the equations were fitted by least squares and the constants are as follows: a=6.151, c=i4, k=O.236 in, m=1.050, n=1.684, p=11.632, s= O-269_ Theoretical predictions for the endoclastic part of the fragments resulting from single fracture claim that rn= 1.0; in the present case the sample consisted of approximately 55 o/o endoclastic, the remainder being exoclastic material_ Equation (7) is satisfied to the second decimal place, since m+n+s=3.003. W& calculated from eqn. (9) for x1 =0_0116 in. and x,=0236 in. is 24.294 g, while the experimental value was 24.728 g. The difference of 0.434 g can be accounted for by the fact that the
1,
0.02
0.03
NOMINAL
0.05 SCREEN
O-1
a2
0.3
INCHES
SIZE. I
Fig- 1. Size and shape factor distributions of fragmmu below 3 mesh for five 1 in. diam. glass spheres crushed by slow compression.
3-mesh point- is slightly displaced towards the coarser region, that is to the right of k (Fig_ 1); it is a common finding that eqn. (3) begins to lose validity in the extreme coarse sizes_ On the other hand. 3c2=0.185 in. (4-mesh) fits exactly onto the line, and IV,_, calculated for x1 =0.0116 in. to xz=0.1S5 in. is 18.584 ,e which differs from the experimental value of 18.545 g by only 0.2°k. W, calculated from eqn. (8) is 25.194 g and from eqn. (9) is 25.377 g_ while the weight of the entire sample was 25.827 g. The approximately half gram difference between the measured and computed values of W, is accounted for partly in the coarse region as just discussed_ and partly in the fine region. The calculated value of tVo is an extrapolation based on the measured range only; differences from the true values are due to deviations from eqns. (3), (4) and (5) outside of the measured sizes. In the present work the shapes of the particles were prismoidal and sphenoidal. Thus. the negative exponent in eqn. (5) indicates that particle shape changes from flaky to more regular forms as the size diminishes_ This was confirmed by microscope measurements. It was also found that particles become more elongated with decreasing particle size; this is a further reason for the negative exponent in eqn. (5). Several investigators have reported that shape factor is a function of particle size’-*, and there is other evidence in the literature supporting the view that shape factor does not remain constant as particle size changesq*lo_ On the other hand, theoretical derivations of size distribution equations based on statistical-mechanical considerations frcquently assume that particleshape remains wnstani with char&g particle size”-‘* ; transformations from number to weight distributions are thus performed with a wnstant shape factor- The self consistent nature of the data presented here supports the view that particle shape can change with size in the relatively warse size region. There is some evidence that changes in the particle shape factor diminish as particle size becomes smaller. Particle shape factors are not measured routinely in wmminution studies, an omission which if remedied could yield useful information_ It is emphasized that the present findings refer only to the relatively coarse size region. Methods are available for particle shape factor determination in the fine sieve range’ (500-50 m), and a method for counting particles in the region from 50 to 1 p can be adapted to provide other data necessary for shape Powder
Technot
2 (1!%53;69) B-60
60
SHORT
COMMUNI
factor determination. Finally, it is stressed that the degree of accuracy achieved in satisfying eqns. (7), (8) and (9) is principally due to the favonrable form of the shape factor distribution curve (Fig. 1) ; other materials and other modes of breakage do not give such little deviation from a straight line. C. C. HARFUS
AXGZ
G.
STAMBOLTZIS
Henry Krunzb School of Mines, Colwnbk University, New York ( U.S.A.)
2 3 4 5 6 7 8 9 10 11 I? I3 14 15
REFERENCES 1 C. C. HARRIS .ahD H. G. S~IH. Second Spnp. Preporafion. Unirersiry of Leeds, 1957. Paper 9.
CATIONS
OR Coal
The importance of temperature control in Conlter conuter dysis Materials described as insoluble usually have a finite solubility. and in particle size analysis, where very dilute suspensions are required it is often necessary to dilute with a saturated solution of the suspended solid. When applying this procedure in this laboratory to ball-milled phenacetin, considerable variations occurred between successive readings on the Coulter Counter, at the same size setting It was observed that a small rise in temperature, brought about by the aperture illumiuatiug lamp, coincided with a fall in the count_ The effect of temperature was therefore investigated further, and extended to other materials. Experimental IlZateriaIs. Phenacetin and acetylsalicylic acid were of B-P. quality and isopropanol of general purpose reagent quality. Samples of griseofulvin and procaine penicillin were obtained from Glaxo La-. boratories Ltd., phenothiazine in aqueous suspension from 1.C.L Ltd, and “Goulac- from Production Chemicals (Rochdale) Ltd. Procedure_ Phenacetin, acetylsalicylic acid, griseoftrlvin and procaine penicillin macrocrystals were size -graded using BS. sieves. A closely graded fraction of each was wet ball-milled in the manner described previouslyl. Nonidet P42 and Dispersal T were added to the mill as wetting and dispersing agents for pheuacetin, Tween 80 for acetylsalicyIic acid and Goulac for griseofirlvin. Procaine penicillin was milled in acetone.
A. 0. GAITS, Trims. AIhfE, 51 (1915) 875. A M_ GAUDW, Trims_ AIhfE, 73 (1926) 253. R ScxnnmAh%v. AIME Tech. PubI. No. 1189.1940. J. J. GILVARRY AhD B. H. BEFK-0x1, 1. A&. PJq-s., 32 (1961) 400. R D. CADPartide Si=e Deferminmion. Interscience. N.r, York, 1955. C. C. HARRIS, Nature, 187 (1960) 401. L. W. NEEDHAS%ASD N. HILL, Fuel. 14 (1935) 222. G. G. BROW, Unit Operarions, Wiley. New York, 1950. U. N. BHRASX AXD J. H_ BROWN. Tams AIhfE. 223 (1962) 248. J. G. BES~XTT. 1. Znx Fuel. 10 (1936) 22 A. B. MAXXISG. J_ INI. FzteI_, 25 (1952) 31_ J. J. GILVARRY. J. Appl. PIyr., 32 (I 961) 39 I_ A. M.GAI_DXSA~DT_ P. MELOY, Trmrr. AIME,t23(1%2)40. R. R. KLDIPEL fihD L. G. AUSITK. Truns. AIhZ&, 232 (1965) 88.
Received December 29, 1967
Determination of solubility. Preliminary experiments showed that solutions of phenacetin in 50% ethanol gave an absorption maximum at 248 rnp which obeyed Beer‘s Law. An excess of phenacetin was heated in water and cooled to tbe required temperatitre- A sample of the clear solution was withdrawn, accurately diluted with 50% ethanol and assayed by measuring its extinction at the maximum. Griseofulvin and phenothiazine were examined in a similar manner using the peaks at 2945 mp and 253 mp respectively. Phenothiazine solubility results were very scattered, probably becanse it was very difficult to wet and therefore required an excessively long time for complete saturation- The solubility was of the order of 1 x 10m3 per cent at WC and 27J°C. Saturated solutions of acetylsalicylic acid in isopropanol were prepared in the same way, but were assayed by evaporating to dryness in a vacuum oven at 80°C and weighing the residue. Procaine penicillin was determined by the iodometric procedure of the British Pharmacopoeia. Aqueous solubilities of acetylsalicylic acid at various temperatures have already been determined2. Results are summarized in Table l_ Particle size determination. A Coulter Counter model A was used with 30-p and 70-p orifice tubes and O-OS-ml and 0_5-ml manometer volumes respectively- The orifice tubes were calibrated in the normal way using monosized pollen grains. Background counts in all the experiments were small The temperature was controlled during analysis by a 150-ml jacketed vessel. Purified water was cirPowder Technol_ 2 (1968/69)
58-60