- Email: [email protected]
The failure properties of lactose and calcium carbonate powders
Powder Technology-Elsevier Sequoia SA., Lausanne-Printed in the Netherlands
329
The Failure Properties of Lactose and Calcium Carbonate Powders SILVA KOtOVA and N_ PILPEL Department of Pharrracv . Chelsea College of Science_ linirersirr of London- .tfanresa Road London_ S_ 11 _ 3 lGr . Briraini (Received March 12 . 1971 : in revised form Au_eust 26. 1971)
Summary An annular shear cell has been employed to investigate the failure properties of taro cohesive materials, a-lactose monohydrate and calcium carbonate, and to classify then by the way in which their fundamental parameters depend upon their state of consolidation_ It has been shown that the two materials are examples of "simple" powders ; the angle of internal friction, A, provides a satisfactory single parameter for describing their properties. A general relationship between A and the particle size of the powders has been established over the sire ranges and stresses investigated .
INTRODUCTION Jenike r -2 and others` extended the basic principles of soil mechanics to describe the flow and failure properties of fine, cohesive powders, on the assumption that they could be treated as modified Coulomb solids . When such powders are consolidated and sheared under an applied normal stress they fail_ By plotting the shear stress at failure against the applied normal stress (or alternatively, against the compound normal stress), one obtains the powdei s yield locusA separate yield locus is obtained for each state of consolidation or packing density of the powder bed_ The end points of the loci are of particular significance, since they represent conditions where the powder is failing in shear without a change in volume occurring . Various equations have been proposed to describe the shapes of the loci'-' -'. In particular, Williams and Birks 6 showed that when the end points fell on a single curve passing through the origin of the coordinate system employed, it was possible to express the failure properties of the powder by means of an angle of internal friction and other fundamental parameters .
If these parameters are independent of the packing density, the powder is described as "simple". but if one or more of them vary with the packing density, the powder is described as "complex_ Relatively little data have so far been published for distinguishing between the two categories 6_ In the present work . shearing tests were carried out in an annular shear cell', instead of the more usual type of Jenike cell', on two different materials x-lactose e monohydrate and calcium carbonate. over a range of particle sizes and packing densities. The results were combined with those obtained in a tensile tester_ The resulting loci were analysed to see whether their shapes conformed to the equations proposed by Williams and Birks 6 •' and whether the failure properties of the materials could therefore be exprecsed in terms of an angle of internal friction and other fundamental parameters . One purpose of the investigation was to assess the operation and performance of the annular shear cell ris-a-ris that of a Jenike cell_ Other objectives were to establish whether the two materials investigated behaved as "simple" or as "complex" powders and to see whether any correlation could be established between the particle sizes of the powders and one or other of their fundamental parameters .
EXPERIMENTAL Materials
The two materials investigated were x-lactose monohydrate (Whey Products Ltd) and precipitated calcium carbonate (Solvay et Cie and J . & E_ Sturge)_ The former was classified into close size fraction on an Alpine airjet siever or a Bahco classifier, the latter on an Alpine classifier (by courtesy of the Warren Spring Laboratory)The lactose was representative of amorphous, organic powders having relatively low particle Ponder Tec noL 5 (1971,772)
330
S_ KOeOVA. N . PILPEL
TABLE I Physical properties and particle sizes of fractions of lactose and CaC0 3 Calcium carbonate
7-Lactose monohrdrate Size range (ltm)
Average size (pm)
Size range (urn)
Average size (!tm)
3-4 4-7 5.5-8 .5 7.5-9.5 12-0-15-0 40-55 Molecular weight Melting point. ° C Particle density . g/cm 3
4.0 5-5 7.0 8.5 135 47 .5 360.3 210-214 (for range 0-50 pm) 1 .550-1 .552 0.01
0-3 3-5 8-12 12-14 14-16 20-30
1_5 40 100 130 150 250 100.9 1339 2.711
Moisture content . °o w,w
density and melting point ; the calcium carbonate was representative of crystalline inorganic powders with a high particle density and melting point . Relevant physical properties of the two materials are given in Table 1 together with the particle size range and sizes of the various fractions . These were determined by optical microscopy_ The fractions were dried at 105 ° C until their moisture contents were ca. 0.01 % (w/w). This level was maintained throughout by performing the experiments in a room fitted with a Drymatic 50 dehumidifier, which kept the ambient relative humidity at a constant value of 60± 5 % . Apparatus and procedure Two pieces of apparatus were used for investigating the failure properties of the materials . (a) The tensile tester (constructed by K . C_ Products) was of the same design as used by Ashton et al_'_ It consisted of a shallow, cylindrical brass dish. 9 cm i .d. and 1 .0 cm deep and was split vertically in half, one half being fixed to the base, the other resting on steel ball bearings which moved in grooves . The powder was introduced into the cell, the two halves being clamped together, and it was then consolidated by twisting the lid ten times through an angle of 20°, under various normal loads . The movable part of the cell was unclamped and an increasing tension was applied to it through a calibrated spring, connected to a shaft moving at 0.125 cm/min. The tensile stress of the powder and the particular packing density concerned were calculated from the time taken for the two halves of the cell to
001
separate, and from the weight and volume of the powder filling it . By applying different consolidation loads, tensile stress and packing density values were obtained, which were fitted to a best straight line by regression analysis. The equation of this regression line was then used to calculate the tensile stress for any value of the packing density achieved during shearing of the powder in the shear cell . (b) The shear cell (kindly constructed by Glaxo Laboratories Ltd .) was of the annular type' and is shown in Fig. 1(a) and diagrammatically in Fig. 1(b). It consisted of an annular shaped trough covered by an annular lid containing flights at 20° intervals. The flight level defined the shear plane between the powder in the lid and that in the cell . The lid was restrained during shearing by a hardened tempered steel bar equipped with strain gauges (silicon semi-conductor types-2A-1A-120P ( + ) and 2A-1A-120N(-) from Ether Ltd .). Both arms of the bar were restrained to minimise tilting of the lid and ensure accurate recording of the torque being exerted on it. A normal load was applied via a platform attached to the shaft of the lid; it was possible to reduce the weight of the lid assembly by means of a counterpoise weight and a pulley (Fig . I)30-70 g of powder (depending on the particle density) were needed to fill the apparatus (permitting a large number of measurements to be made on relatively small samples) . But because of the design, the shear strain varied across the width (r_-r 1) of the shearing area . This meant that the powder at the outer wall was being sheared more than that at the inner wall . However, errors were minimised by Ponder Technol_ 5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS
making (r 2 -r 1 ) small and by employing a low rate . of shear of I rev/hour The torque developed during shearing was measured from the output of the strain gauges. These formed four arms of a Wheatstone bridge which %vas connected to a pen recorder and chart (Heath Co.) of variable speed and sensitivity. (The connections to the recorder appear on the left in Fig- I(a)_) To prepare a sample for testing the material was introduced into the cell in thin layers, packing it slightly with a light disk- Care -vas taken to ensure uniform height of the bed throughout the cell and thus prevent tilting of the lid. The lid was then lowered on top of the powder and a chosen load Was applied normally . By twisting the lid a predetermined number of times through 20', the powder was preconsolidated to a certain packing density, which could be varied by changing the applied normal load. Although the geometry of the apparatus allowed
331
for unlimited movement during shearing (in contrast to that of the Jcnike apparatus) . the procedure was to minimise the movement by consolidating the sample nearly to its critical state . before shearing it to failure. Consolidation --as completed in the next stageby causing the sample to fail under a given stress until a steady state was reached, or closely approached. This was achieved in the following -vat •_ With the restraining bracket fixed over the steel bar, the motor --as switched on in order to rotate the turn table and the output of the strain gauges was recorded_ The rotation was stopped when the chart reading levelled off (after an initial maximum)_ showing that the sample was then flowing without further increase in the shear stress . Provided that the powder had not become compacted during the consolidation process . the shear force was removed by rotating the lid back by hand, Powder TechnaL 5 (19=1 72)
S. KOCOVA, N. PILPEL
332 the load was reduced and the sample was then again sheared to failure under this reduced load . The normal stress and the shear stress at failure gave one point on the yield locus of the powder at the packing density obtained during preconsolidation of the sampleThe bulk density of the sample was calculated from the volume and weight of powder in the shear cell when full, and the packing density by dividing the bulk density by the particle density of the material- In general a fresh, unsheared sample of powder was employed for the determination of each yield locus. Assuming the shear stress has developed uniformly over the annular area of the shear cell, the total measured torque O_ is given by 8
0 Sym'xi o X
e a 40
Packing density 0336 0380 a414 a455 50
60
Normal stress, 6„ (9/cm 2 ) Fig. 2(a). Yield loci for a-lactose monohydrate. a%crage size 7_0 Pro-
r_
Q =
r,
rr 2 7rdr
Q = 3R -r(r3-ri)
or
(1)
where z is the shear stress, g/cm 2 r) is the inner radius of the cell, cm r, is the outer radius of the cell, cm . By applying known torques to the apparatus (by hanging weights over the system of horizontal and vertical pulleys) and noting the readings on the recorder chart, it was possible to convert the readings into values of shear stress T by means of the above equation (1). Relatively little data are available on the shear behaviour of lactose and CaCO 3 t o and it was not possible to make a direct comparison between the present results and those that would have been obtained in a Jenike cell because of the smallness of the fractionated samples available. However, measurements made in the annular cell on a sample of titanium dioxide gave a value for A within 2° of that obtained' in a Jenike cell . Measurements in an annular cell tend to veer on the side of safety in regard to predictions about critical blocking of apertures for hoppers and bunkers' . Other advantages of the annular cell over the Jenike cell were that it enabled one to employ relatively small amounts of powder and considerably simplify the procedures of consolidating and shearing the samples. RESULTS
The experimental results and the parameters derived from them are given in full in Tables 2 and 3 .
20 30 40 50 Compound normal stress
4+T (g/cm 2 ) Fig. 2(b). Yield loci for a-lactose monohydrate, average size 7.0 pm, using compound normal stress . Figures 2(a) and 3(a) show the yield loci of two representative samples plotted in the usual way with the normal stress aN, g/cm2 as the abscissa and the shear stress at failure, r, g/cm 2 as the ordinate_ The line joining the end points of the loci to the origin is the critical compaction yield locus and its slope is denoted 6'. Figures 2(b) and 3(b) show the corresponding modified yield loci, which were obtained by plotting the shear stress at failure against the compound normal stress (aN +T) or (a3), g(cm2. The line joining the end points of the modified loci to the origin is the internal friction yield locus and its slope, A, is the angle of internal friction of the material for sustained yield . In Figs. 4 and 5 all the yield loci have been reduced to a series of parallel straight lines by plotting log T against log (aN +T) (or log aN)_ It was not possible to obtain sufficient experimental points at any particular packing density to justify fitting each line by regression analysis . But since the original loci at Powder TeehnoL, 5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS
333
TABLE 2 Experimental results and derived parameters . a-Lactose monohvdrate Tensile stress.
Packing
Applied
density,
normal
T
pa/pp
stress_ ~c_N
(g/~')
(gfcm - ) Arerage size
Compound normal
stress. cti (9/c"2 )
r (g: czn9
Reduced
Reduced
normal
compound normal
stress e.,'GE
Reduced shear stress, r'CE
stress.
Reduced
compound shear sties . r'eE
G~'r'E
4.0 Ian
1-05 0-152 265 3 .70 1.05 0.152 4.05 5-10 1-05 0-152 7_20 8.25 1-05 0-152 8-85 9-90 232 0-172 2-65 4-97 232 0-172 6-15 8 .47 232 .172 0 12.25 1457 2.32 0-172 17-42 19-74 3.65 0222 1150 15-15 3-65 0.22 21 .90 18-25 3-65 0.222 26.10 29.75 0-)-)-) 3-65 37-85 3420 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S, deg Angle of internal friction, deg Deviation from reduced stresses curie Deviation from linearised yield loci Average
Shear stress at failure
6.25 6-75 920 11-10 9.2-1 11 .80 17-21 20-00 20-05 2325 30-30 38-60
0300 0-457 0.815 I-000 0.152 0353 0-703 1.000 0336 0535 0-763 1-000
0374 0-705 0.515 0.762 .835 0 1-040 1 .000 1250 0-252 0.530 0 0-677 .429 0.738 0-990 1-000 1 .146 L402 0588 0583 0-681 0.790 0.880 1-130 1-000 p =03722-2-95 P' = 0338=33 0' .124_73% 4 =0 q° = 0.110=6.3 n = 1 .87 _23 6' =520 A = 49.0 7-0 4.3
0.625 .682 0 0-930 1-120 0.467 0.59S .872 0 1-012 0-532 0-615 0.794 1.020
4.25 8.80 10.75 12.70 7.20 9.50 13 .20 16.10 20.80 18 .00 23.60 3020 3755 40.60
0.076 0.523 0687 I .OOC 0-038 0-134 0-309 0-512 1 .000 0271 0-505 0.750 1-000 0506 0-670 0-830 1-000
0.115 0-495 0.560 1-023 0.703 1.250 1 .000 1 .478 0.117 .428 0 0205 0.565 0366 0.785 0-552 0-960 1 .000 1-240 0314 0567 0535 0544 .765 0 0-950 I-000 1-180 0534 0-864 0-688 .985 0 0-840 1-085 Low 1-175 p =0-320=&7% P° =0298_14-0% q =0.060-5-0 °e q° = 0567 _-5.8 01 n =1 .81 _20% a' =50 A = 48 6.0 5.2
0-467 .967 0 I-ISO 1395 0393 0.520 0-722 0 .880 1-136 0 .533 0=7(10 .895 0 1 .110 .815 0 0-930 1-025 1-110
size 5.5 lnn
0.109 050 0.65 1 .15 0-50 0.109 450 5.10 0.50 0.109 5.90 6 .40 050 0-109 8-60 9.10 1 .50 0212 0-65 2.15 1 .50 0212 225 3.75 1 .50 0212 5.20 6.70 1.50 0.212 .60 8 10.10 1.50 0.212 16.80 18.30 0280 &60 10.60 2.00 200 0280 16.00 18-00 200 0280 23.80 25-80 200 0280 31 .75 33 .75 275 0.345 23.80 26.55 0345 275 3150 3425 275 0.345 41.75 39.00 275 0345 47.00 492 :5 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) S hear index Angle S', deg Angle of internal friction, deg 7 Deviation from reduced stresses curve Deviation from linearised yield loci
4625
51 .05 55-25
(conrinued) Powder Tech oL. 5(197 ;/72)
S. KOCOVA, N . PILPEL
334 TABLE 2 (continued) Tensile stress, T (g/cm 2)
Packing density pB/pp
Applied normal stress, cN (g/cm2)
Compound normal stress, ar (9/pn2 )
Shear stress at failure, T
Reduced normal stress, aN/cE
Reduced compound normal stress,
Reduced shear stress, VICE
Reduced compound shear stress, =/GE
aV a;
(g/cm 2 )
Average size 7.0 Etm 0.50 0221 1.75 2.25 0.50 0221 325 3 .75 0.50 0.221 4.95 5.45 0.50 0-221 6.35 6.85 0.50 0221 7.90 8.45 1 .00 0.250 1.75 2.75 .00 1 0250 4.95 595 1 .00 0.250 7.90 6.90 1.00 0.250 10.95 11 .95 1_00 0250 16.20 17.20 0-272 1.75 3_75 2.00 2.00 0272 4.95 6.95 7.90 200 0272 9.90 200 0.272 10.95 1295 2.00 0.272 16.20 18.20 200 0.272 23.45 25.45 200 0.272 31.00 33.00 245 0298 23.45 25.90 245 0.298 31.00 33.45 245 0298 38.75 4120 245 0.298 44.45 4200 245 0 298 4620 48.65 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle b', deg Angle of internal friction, deg % Deviation from reduced stresses curve Deviation from linearised yield loci
330 3 .65 4 80 5.85 7_50 4.80 7.25 9.20 11 .25 15.00 8-20 10.25 13.10 14.50 1800 23.00 2930 3020 35.00 39_50 41 .50 43.50
02 2 2 0.412 0 627 0.805 1.000 0.108 0306 0.488 0.675 1_000 0.056 0.159 0.255 0353 0_523 0.756 1 .000 0310 0 671 0.839 0.910 1 .000
0268 .418 0 0.462 0.443 0.645 0.608 0.810 0.740 L000 0-950 .160 0 0296 0346 0.447 0.400 0.567 0.695 0.695 1 .000 .925 0 0_114 0265 0210 0331 0300 0.423 0393 0.468 0.552 0.581 0.772 0.740 1 .000 0.945 0.655 0-532 0.688 0.757 0.845 0.856 0.914 0.898 1 .000 0.941 p =0_191±26% P'=0.724±22 q =0.066±10.6% q° = 0.057-1-103 n =1_72±3 .0% 6' = 43 A =42 3.0 3.0
0390 0_432 0-568 0.692 .886 0 0279 0-421 .535 0 0_655 0.872 0248 0311 0397 0.440 0.545 0.697 0.890 0.621 0.720 0.812 0.853 0.895
625 8.00 9.45 9.75 10.50 10.75 7_65 9.20 11.70 14.05 15.90 7.60 10.30 1250 14.80 17.50 12.50 16.40
0.179 0337 0.442 0.668 0.843 1.000 .111 0 0.308 0535 0_726 1.000 0-10D 0.280 OA70 .660 0 1.000 0.158 0367
0-291 0_427 0518 0-705 0963 1 .000 0224 0.397 0.595 0 .763 1,000 0224 0378 0.543 0.705 1 .000 0.203 .415 0
0.568 0_727 0.859 Q886 0.955 .978 0 0.435 0.521 0664 0796 .905 0 0386 .523 0 0.635 Q751 0.891 0366 .480 0 (continued)
Average size 8 .5 pm 1.50 150 1.50 1.50 1.50 1.50 225 225 225 225 225 270 2.70 270 2.70 270 4.20 420
0384 0384 0384 0384 0.384 0384 0392 0392 0392 0392 0392 0_416 0.416 0.416 0.416 0_416 0.429 0.429
1.70 320 4.20 6.25 8.00 9.50 1_70 4.75 825 11.20 15.40 1.70 4.75 8.00 11.20 17,00 4.75 11 .00
320 4_70 5.70 7_75 9.50 11_50 395 7.00 1050 13.45 17.65 4.40 7.40 10.70 13.90 19.70 895 1420
0_658 0.842 0995 1.026 1.105 1_132 0-497 0.598 0.760 0915 1.030 0-447 0.605 0.735 0-871 1.030 0.417 0547
Powder TechnoL,5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS
335
TABLE 2 (continued) Tensile stress_ T (g/cm2)
Packing density Pe/PP
Applied normal stress,
Compound normal stress,
Shear stress at
Reduced normal
failure,
stress,
Reduced compound normal
GN
GH
7
Gy/GE
stress,
(q/cm2)
(9/cm -)
(g/cm2)
420 0.429 15.65 19-85 420 0.429 30-00 3420 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S', deg Angle of internal friction, deg Deviation from reduced stresses curve Deviation from linearised yield loci
z;GE
Reduced compound shear stress, T., GE
fir/GE
2030 30 .75
0.522 1-000
3.87 6-28 7-61 4.45 6-05 850 10-21 11-73 13-10 635 820 1028 11-87 14-58 2032 2250 24-41 1321 1435 17.84 27.68 3832 49-61
0211 0.809 1.000 0.109 0301 0-417 0-686 0.885 1 .000 0.056 0-156 0268 0355 0517 0-755 0.915 1.000 0.078 0.133 .258 0 .500 0 0.746 1 .000
7
Arerage size 135
Reduced shear stress,
0-580 0-676 1 .000 1-025 p = 0-39517-6 °e p- = 0334-10-6 %e q =0-351 x-73 %e q° =01324-3 .6%. n = 155 -L4-0% X = 445 A = 41.5 8-0 5-7
0-594 0.900
gm
0-19 0250 1-75 1.94 0-19 0250 6-72 6.91 .19 0 0250 831 850 0.35 0.274 1.75 2.10 035 0274 4.85 520 035 0274 6.72 7-07 035 0274 11.05 11 .40 035 0-274 1422 14-57 035 0274 16.10 16-45 0-68 0282 1 .75 2.43 0.68 0282 4.85 5.53 .68 0 0282 831 899 0.68 0.282 11 .05 11 .73 0.68 0282 16.10 16.78 0.68 0282 23.50 24.18 0.68 0.282 28.48 29.16 0-68 0282 31 .17 31 .85 1 .40 0350 4.85 6-25 1-40 0350 830 9.70 1.40 0350 16.10 17.50 1.40 0350 31 .17 3257 1.40 0350 4650 47-90 1 .40 0350 6235 63.75 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile su~ (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S, deg Angle cf internal friction, deg Deviation from reduced stresses curve %Deviation from linearised yield loci
different packing densities were geometrically similar in shape, it was considered legitimate to derive a common slope from all the results, i.e. 1/n,- where n is the shear index of the material, and then fit
0-228 0-466 0.814 0-755 1-000 0915 0271 0-128 Q316 0368 0-430 0-528 .693 0 0-635 0-886 0-729 1 .000 0 .813 0-076 0204 0-174 0264 0_.82 0331 0-369 0.382 0-527 0.4f-, 0.760 0-652 .72 0-915 0 1-000 0-785 0-098 0.212 .152 0 0-230 0 0.274 .286 0510 0.445 0-752 0-615 1-000 .795 0 P =0-148149 p" = 0-145-2-5% q =0-022=45 q~ =0-022-4-0% n = 1_.8 =6 .0 S' = 383 A = 3825 30 9.8
0-355 0-739 0-895 0-271 0368 0-517 0b2D 0-714 0.795 .199 0 .257 0 032 2 0373 0-458 0-638 0-706 0-768 0.2 208 0-)-) 5 0280 0.435 ObO5 0-778
individual lines, with this slope, through the experimental points appropriate to each packing density. The stan dard deviation of the points about these lines was calculated and the figures given at the end Powder Tecimoh 5 (1971172)
S. KOCOVA, N. PILPEL
336 TABLE 3 Experimental results and derived parameters . Calcium carbonate Tensile stress. T (g/cm2)
Packing density, ps lpp
Applied normal stress,
Compound normal stress,
a-%
c,
(g/cm2 )
(9/cm 2 )
Reduced applied normal stress,
Reduced compound normal stress,
e-dal.
e.%/ee
1 .80 525 830 4 .90 9.01 11-15 1428 1732 18 .02 10.71 13-05 17-50 25.75 15.20 23-98 28.81 33 .01 34-05 19-23 3272 5250 59.05 59.50
0-204 0-627 1 .000 0.125 0309 0-444 0-694 0.875 1 .000 0.216 0357 0-559 1 .000 0-267 0.514 0.710 0.949 1 .000 0-147 0-570 0-850 0 963 1 .000
0232 0.238 0-648 0.677 1 .000 1-071 0.164 0303 0340 0.557 0512 0.689 0-708 0.883 0-880 1 .070 1 .000 1 .113 0.250 0-462 0385 0563 0578 0-755 1 .000 1 .111 0300 0.490 0536 0-772 0.723 0-928 0 .952 1-063 1 .000 1 .097 0.183 0353 0.589 0.601 0.964 0-957 0.964 1 .084 1 .093 1 .000 p =0.23±9.4% p" = 0214±9-8 q = 0.046±5-0 a" = 0.044 x-4.5 n =1 .48±25 0//. S' = 48.0 A = 47.5 3 .4 3 .27
0Z 0.648 1 .025 0.289 0-532 0.659 0.843 1 .023 1 .064 0.442 0539 0-722 1 .063 0.467 0.737 0-885 1 .014 1 .046 0.338 0-575 0.923 1-038 1-046
1265 13 .70 17.43 18 .05 13-73 18 .03 22.00 23.58 24-68 15-21 21 .50 33 .05 35.45 35 .65 39.73
0.515 0.624 0.821 1 .000 0341 0540 0.795 0.924 1 .000 0.253 .492 0 0.839 0.931 1 .000 0.404
0.523 0.671 0.822 1 .000 0352 0.576 0-799 0.925 1 .000 0.265 0.500 0.841 0.934 1 .000 .415 0
0.792 0-860 1_090 1 .130 0570 .749 0 0911 0976 1 .021 0-469 .661 0 1 .018 1-091 1.095 .731 0 (continued)
Shear stress at failure r (g/em')
Reduced shear stress, r/cE
Reduced compound shear stress, z/c;
Arcrage size 1 .5 pm 035 0.235 1 .58 1 .93 4.73 035 0.235 5.08 7.75 8 .10 0.35 0.235 0.75 0-245 203 2.78 5.00 5.75 0.75 0.245 7.91 0 75 0.245 8-66 0.75 0.245 1123 11 .98 0 .75 0 .245 14 .15 14.90 0.75 0.245 16 .18 15 .93 1 .05 0-257 5.00 6 05 1-05 0.257 8 .28 9 .33 1 .05 12 .95 14 .00 0.257 1 .05 0-257 23 .28 24-23 1 .50 828 0.274 9.78 1 .50 0274 15.95 17 .45 1 .50 0.274 2103 23 .53 1 .50 0274 29.48 30-98 0274 150 31 .05 32.55 0350 37.98 10_43 2.45 245 0350 31-05 3350 2-45 0350 46.30 48.75 245 0350 5241 54 .86 2.45 0350 54-45 56-90 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle 8', deg Angle of internal friction, deg Deviation from reduced stresses curse Deviation from linearised yield loci
Arerage size 4.0 lire 0.25 .25 0 0-25 025 0.38 0.38 0.38 0.33 038 052 052 052 052 0.52 0.71
0-178 0.178 0-178 0.178 0236 0.236 0.236 0.236 0.236 0351 0351 0351 0351 0351 0.419
8 .10 9.28 12 .91 15.74 8-10 1281 18.90 21 .95 23.78 8 .10 15.74 26.83 29.78 3200 15 .74
8.35 1007 13.16 15.99 8-48 13.19 19-28 2233 24.16 8 .62 16.26 2735 30.30 32.52 16 .45
0.805 0-871 1 .110 1 .145 0.580 0 .757 0.925 0.992 1 .037 0.476 0.671 1 .033 1 .110 1 .113 .745 0
Powder Teehnol, 5 (1971/72)
337
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS TABLE
3
(continued)
Tensile stress, T (4/c"12 ) 0.71 0-71
Packing density, pa/p,,
Applied normal mess, UN
(g/cm2) 0.419 0.419
37.26 39.02
Compound normal mess, a;
Shear stress at failure t,, ~
Reduced applied normal stress,
Reduced compound normal stress,
Reduced shear mess, TIC,
(4/CM2)
(&m2 )
G\IGE
GS/GE
37_97 39.73
41 .03 41 .20
0955 . 1 .000
0_974 1-050 1-000 1-052 p =0247±8 .%o Pr =0243-8 .2%0 q =0-0167--+-8-9% q' = .0163 0 -r 92 ie n = 1 .84 _ 2 .9 ;{ v' = 47.0 A = 46.4
Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S, deg Angle of internal friction, deg Deviation from reduced stresses curve Deviation from linearised yield loci
Reduced compound shear stress. TJeE 1 .030 1 .035
4.0 27
Average size 10_0 fun 035 0.25 0.25 .25 0 040 0.40 0.40 .40 0 055 055 055 0_55 0_55
0292 0-"9 0292 0292 0324 0324 0324 0324 0363 0.363 0363 0363 0363
4.95 732 14.05 1521 7.82 15.25 2132 2355 7.82 15 .48 23 .05 2950 31_05
5-20 8_07 1430 15.46 820 15_65 21 .72 23.95 837 16 .03 23 .60 30-05 31_60
4-23 735 1202 12.52 7.80 13_81 18.95 2020 9.25 1520 20-00 26.70 2750
0326 0514 0.924 1 .000 0333 .649 0 0 .905 1 .000 0252 0-499 0.742 .950 0 1_000
Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S', deg Angle of internal friction, deg Deviation from reduced stresses curve Deviation from linearised yield loci Average size 015 . 0.15 0.15 025 0.25 025 030 030 030 030 030
13.0
0336 0-278 0.477 0.52 7 .925 0 .790 0 0323 1-000 0342 0332 0 .588 0 .653 0.92 7 .805 0 1 .000 0_858 0265 0,298 0507 (1490 .644 0 0.747 0-951 0350 1_000 _870 p =0_083-6 .0% P7=0-082-z61 %e q =0 .01715.9% q"=0-0167 -14-2% n =1 .11=3 .7/.
0-2.74 0.469 0.777 0.810 0326 0377 0.791 .843 0 0.2293 0-481 0-633 .845 0 0.854
3' = 44-0 A = 433 2.6 29
fun
0174 . 0.174 0_174 .191 0 0.191 .191 0 0.211 0.211 0211 0211 0211
7.84 14.05 15.72 7.85 21 .42 2522 735 15.72 2322 29.65 31 .18
8_00 1420 15_87 8 .10 21 .67 23 .47 8 .15 16_02 2352 29.95 31 .48
Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress)
9.15 1320
0.499 0394
14.00
1.000
9.75 17.65 18.00 1030 1525 2132 26 .00 2625
0338 .922 0 1 .000 0252 .504 0 0-745 0.950 1 .000
0-582 0.504 0.895 0340 1 .000 0-891 0-345 0_420 .923 0 .760 0 1 .000 .775 0 0259 0330 .509 0 .489 0 .747 0 .684 0 .951 0 0834 1 .000 0.842 p =0.133-3A% .132=3.0%'o P- =0 q =a10-=9-8%
0.577 .832 0 0282 0-415 0_752 0-767 0327 Q484 Q677 0226 .834 0
q" =0.009=8.1 %o
(contin ud)
Powder iechnol, 5
(1971172)
338
S. KOl° OVA, N. PILPEL
TABLE 3 (continued)
Tensile stress, T
Packing density, Ps/Pr
(9/tent)
Applied normal
Compound normal
Shear
stress, ar. (gfc n) '
stress,
failure
Reduced applied normal
Cr
=
stress,
compound normal stress,
(g/cm2)
(9/c"12 )
CN/CE
Cr/CE
stress at
Shear index Angle S', deg Angle of internal friction, deg Deviation from reduced stresses cure Deviation from linearised yield loci
Reduced
Reduced shear stress, T/cE
Reduced compound shear stress,
T/cE
n =1.35-25% 6' = 39.8 A = 39.5 3.0 3_1
Arerage size 15 .0 pan
0_15 0363 325 3.40 0_15 6 .05 0363 6.20 0.15 0.363 7_90 8_05 0.42 0_413 325 3_67 0.42 0_413 7.90 832 0_42 0_413 14.75 15_17 0.42 0.413 16.32 16.74 0_42 0.413 =42 2200 0.42 0_413 23_72 24.14 0_75 16 .32 0.480 17.07 0.75 0.480 23.72 24.47 0_480 0.75 31-78 32-53 0.75 0.480 37.95 38.60 0.75 0_480 39.75 40.50 0_95 0.706 23.72 24.67 0.95 0.706 31 .78 3273 0.95 0.706 39.75 40_70 0.95 0.706 4730 48.25 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile strss (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S', deg Angle of internal friction, deg Deviation from reduced stresses curve Deviation from linearised yield loci
1.92 4.05 4.75 2.95 6.55 9_50 11.25 13_76 14.45 14 .50 18.52 21.05 25.00 25.55 19.82 2'-)65 27.75 31.05
of Tables 2 and 3 are the maximum percentage deviations about the mean values. In Figs. 6 and 7 all the points from the yield loci have been replotted on axes of reduced shear stress TR versus reduced applied normal stress c R (where rR = T/cE and cR = cN/cE and cE is the applied normal stress at the end point of the appropriate yield locus). It is seen that for each material, the points fall very reasonably on a single curve (fitted by the method of non-linear correlation) ; the percentage deviation of the points is given in Tables 2 and 3 . Figures 8 and 9 show the straight lines to which the yield surfaces are reduced when the logarithm of the reduced compound shear stress (TR=T/(E E ± T)
0.411 0.766 1.000 0.137 0333 0_622 0_688 0_927 1.000 0_411 7597 0.799 0.955 1 .000 0.501 0.672 0.840 1 .000
0.422 0-243 0.770 0.513 1 .000 0.601 0.152 0.124 0345 0.276 0401 0 628 0.693 0.474 0.929 0_580 1_000 0_609 0_421 0365 0_604 0.466 0.803 0.530 0.953 0.629 1 .000 0.643 0.511 0.419 0_678 0_479 0.844 0.587 1_000 0656 p =0.118±20% p'=0.116±1 .7%
0239 0.503 0590 0.122 0.271 0.394 0.466 0_570 0.599 0358 0_457 0.520 0.617 0.631 0_411 0.469 0.575 0_644
q =0.019±53% q'=0.0183±7.6% n = 1-23-+-3.0% 6' = 31 .0 A = 31 .0 8_0 4.6
is plotted against the logarithm of the reduced compound normal stress (cR=(cr,-i-T)/(cud-T)) . In Figs. 10 and 11 the values of tan A for the different fractions of lactose and calcium carbonate have been plotted against their average particle size in order to establish a relationship between the angles of internal friction of the materials and their particle size. DISCUSSION
It is evident from Figs . 2(a), 3(a), 4 and 5 and also from the results in Tables 2 and 3 that the yield loci Powder TechnoL, 5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS
339
100
15200
S . I.® N
Pocking density Symbol o 0235 x 0.245 0257 a 0 0274 148` 0 0350 1 , I I ∎ I -10 10 20 30 40 50 60 Normal stress, 6. (g/cm° ) 20
1 v 50
P 20
o x c 0
a N 05
Fig 3(a)_ Yield loci for calcium carbonate, average size 1_5 µm .
02 02
+
05
0336 0380 OA14 0455
, 1 I + I 1 1 1 1 10 20 5A IL L 200 500 100 200 500 Compound normal stress, 6u+T(WWcmt)
Fig 4. Logarithm of shear stress at failure as logarithm of compound normal stress for x-lactose monoh3drat- average size 7.0 {tm_
10ao -
soo60_04aoM
, 6„+T (g/cm5 Yield loci for calcium carbonate, average size 1 .5 lam, Fig. 3(b). using compound normal stress 10
20 30 40 50 Compound normal stress,
60
E - 20n rn 1^ h 740L n 60. S 4oL N
of all the samples investigates satisfactorily obey the Warren Spring equation 3 r ° 6N +T (2) CC
T
This equation employs the compound normal stress (6N + T) (as illustrated in Figs_ 2(b) and 3(b) instead of 0N, for deriving information about the failure properties of a powder- The advantage of using this modified yield locus is that it allows for the tensile stress of the powder, as a component of the total normal stress during shearingWilliams and Birks6 •' concluded from theoretical considerations that it is only at the end point of a modified yield locus (designated by the subscript E) that all the work being done by the shearing force is being spent in overcoming the forces of internal friction within the powder. At all other points work is being done against both internal and external
forces, due to changes in the volume of the powder
Symbol x A d
2.0-
.
Packing density 0245 0257 0274 0350
.0 .0 1000 2.0 40 60 10 ZOO Ono 60 Compound normal stress, 64 = w~ 1 (g/~nz) Fig_ 5. Logarithm of shear stress at failure rs logarithm of compound normal stress for calcium carbonate, average size 15 fun 1.0
bed_ They therefore selected this end point as more characteristic of a powder than any other point on the locusEquation (2) does not give any information about the end point of a yield locus or predict that an end point should occur- This is contrary to what has been shown by Jeniker_ For the end points, eqn_ (2) can be rewritten C° _ zE (3) T a; and in cases, as in the present work, Figs 2(b) and 3(b), where the end points of the modified loci lie Powder Technol., 5 (1971/72)
S. KOCOVA, N- PILPEL 1.0
0
Symbol o x A O L
-02
Packing density 0336 0.360 Q414 0 .455
02 .4 0 O5 0.8 Reduced applied normal stress
1
Fig_ 6 . Yield loci for a-lactose monohydrate, average size 7A fan, using reduced stresses .
0]
0.2 0.4 06 08 to Reduced compound stress, (6,•+ T)/(?E+T)
Fig. 8. Linearisation of family of yield hydrate, average size 7.0 fun.
loci for a-lactose
mono-
2.Or
QiO/
10 0.6
VW F-
()S
N H
a
L /1.~w
07
N V
02
O
rN 0.3
t , a 0
02
7
Symbol o x a 0
b d v u 02
0.4
OS OS Reduced applied normal stress, V(SE
•
Fig 7. Yield loci for calcium carbonate, average size 1 .5 fun, using reduced stresses. W .1 0
on a single line through the origin (the internal friction yield locus), one can also write : TE = tan A(C + T) = tan AaE (4) where A is the angle of internal friction of the powder for sustained yield and is a characteristic property of it . Combination of eqns. (2), (3) and (4) then gives' : T = tan A UN(11n) (TE(x-1)/n (5) In view of the unique properties of the end points of the yield loci, it is advantageous to use them in order to calculate the so-called reduced stresses of powders undergoing shear . These are dimensionless
Packing density 0.235 0.245 0.257 0.274 0.350
, Q4 0.6 O.B to Reduced compound stress, O, rtC tT)f(6E +T) 02
2.0
Fig 9. Linearisation of family of yield loci for calcium carbonate, average size 1 .5 fan.
E
quantities' and can be substituted into eqn . (5) to give
1j"
r =tan A cR (6) Here TR is the reduced compound shear stress (as defined in the List of Symbols) and cR is the reduced compound normal stress. By inspecting families of yield loci at different packing densities, Williams and Birks came to the conclusion that for two materials at least, Ti0 2 Powder Tecin+oh 5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS
that have been employed . In these circumstances, one can further simplify the presentation of the yield loci (Le- linearise the yield surfaces) by expressing eqn. (6) in logarithmic form
tz0-
v O
X
O ;, 1D0 xx
~090 I 0
c
\x
x
a
a 0so 05 0 • 5.0 10.0 15.0 20 .0 25 .00 Particle size, t. (jam)
Fig. 10- The effect of particle for a-lactose monohydrate.
•
341
5.0
tao
1s0
Particle
size on
20.0
30.0
500
the angle of internal friction
• 250
30 .0
size Dp (pm) Fig. 11 . The effect of particle size on the angle of internal friction for calcium carbonate.
and penicillin, the cohesion and the tensile stress were both linear functions of a E and aE. This enabled them to define two quantities q (alternatively q") and p (alternatively p"), called respectively the specific tensile strength and the specific cohesion. The same conclusion is now arrived at concerning the samples of lactose and calcium carbonate that have been investigated in the present work- This is shown by the relative constancy (despite a certain scatter) of the values of p, p", q and q" listed in Tables 2 and 3 and also in Figs . 6 and 7, where the reduced loci intersect the ordinate at p and the abscissa at q. The deviation of the points from these loci is about 3-4% (not more than 10% in the worst cases, see Tables 2 and 3), which is considered very satisfactory, bearing in mind the nature of the correlations
log rR = log tan A i I log cg (7) n A straight line of slope 1/n should be obtained by plotting log rR versus log 7R. This is confirmed for both lactose and calcium carbonate in Figs 8 and 9_ Powders for which the quantities A, q", p" and n are independent of the packing density may be termed "simple powders", while those for which they vary with packing density are called "complex"'On this definition, titanium dioxide, pencillin, lactose and calcium carbonate are all examples of simple powders over the size ranges and stresses that have been investigated_ The determination of A, in which c5. is replaced by (cr ~-T), involves a complete analysis of all the stresses that are acting in the plane of shear of the powder. Unlike other quantities, such as C and T, A is independent of the packing density of the powder_ Although the quantity n is also independent of the packing density, A is the only parameter that relates to failure of the powder under steady state conditions, that is, at the end point of the yield locus, where no change in volume is occurring . (in passing, we consider it important that the locus should be fitted to pass through the point T_ Though its determination involves experiments of a different type to those in a shear cell, with the possibility of producing errors in the values of n, the only way of independently confirming the validity of the T measurement would be to obtain other values of -r at negative values of a. Such measurements are not, at present, possible_) A is therefore likely to be a useful parameter for assessing the failure properties of powders. It is also likely to be useful in practical matters, such as the design of hoppers, calculation of critical blocking apertures, etc_ where steady state conditions are obtained during flow. If; therefore, it is concluded that A is a satisfactory single parameter for assessing the failure properties of simple powders, it is of interest to see not only how A varies from one powder to another but also, for example, how it varies with the particle size of the powderIn Figs . 10 and 11 the values of tan A for the different fractions of lactose and calcium carbonate Powder Tecimol . 5 (1971x72)
S. KOCOVA, N. PILPEL
342 have been plotted against their average particle sizes- The two curves are seen to be geometrically similar. Within the range of sizes examined, they have the general form tan A = K+ S exp (-*Dp) (8) where DP is the average particle size of the fraction and K, S and yi are constants . These constants are evaluated by subjecting the curves to treatment on an analogue computer. It is found that for lactose (4Eunn < D P < 50 pm) tan A = 0.77+0.47 exp (-0.019 Dp) (9) and for calcium carbonate (1 .5 . un < DP < 30 ,um) (10) tan A=0 .46+0.64exp(-0.0031Dp) It seems probable that the general equation (8) will also be found to apply to other powders over particular ranges of their particle size provided that they conform to the definition of "simple" powders in which A is independent of the packing density of the powder. CONCLUSIONS
(1) The annular shear cell gives reproducible results for determining the yield loci of powders over a range of particle sizes and packing densities . Measurements can be made on relatively small samples of powder, more rapidly and conveniently than in a Jenike cell . (2) The failure properties of lactose and calcium carbonate have been expressed in terms of four fundamental parameters. These are the angle of internal friction A, the specific tensile stress q", the specific cohesion p" and the shear index n . Since A, q", p" and n have been shown to be independent of the powders' packing densities over the size ranges and stresses investigated, the two materials are examples of "simple" powders . (3) A provides a convenient parameter for specifying the failure properties of "simple" powders . Relationships have been established between A and the average particle sizes of the lactose and calcium carbonate, which are of the form tan A = K + s exp(-%Dp).
sion for helpful discussions and the Directors of I.C.I . for financial support. We also thank Glaxo Laboratories and the Warren Spring Laboratory for the gift of apparatus and materials . LIST OF SYMBOLS
C=C(pB) cohesion, g/cm 2 DP representative particle d iameter. pm n shear index, the exponent of the powder's yield locus equation specific cohesion of the powder (applied P=CICE normal stress) p =` C/ci specific cohesion of the powder (compound normal stress) q=T/cE specific tensile stress (applied normal stress) q" = TIC; specific tensile stress (compound normal stress) rl inner radius of shear cell, cm r2 outer radius of shear cell, cm T=T(p B) tensile stress, g/cm 2 b' angie of inclination of the locus joining the end points of family of yield loci when applied normal stress is used A angle of internal friction of the powder constants K, bulk density, gfcm 3 PB particle density, gcm 3 PP applied normal stress corresponding CE to end point of powder's yield locus . cE=cE +T compound normal stress at end point of yield locus applied normal stress, g/cm 2 UN cr=cN +Tcompound normal stress, g/cm 2 aR=CN/CE reduced applied normal stress cR=cr/aj reduced compound normal stress T shear stress at failure, g/cm 2 shear stress at end point of yield locus TE reduced shear stress (applied normal TR = T/GE stress) reduced shear stress (compound normal TR=TIcE stress)
REFERENCES ACKNOWLEDGEMENT
We wish to thank I .C.I. Pharmaceuticals Divi-
I A. W. Jenike, BuIL 108, Eng. ExpL Sta., Utah State Univ ., 1961. 2 A. W. Jenike, Bull. 123, Eng Expt . Sta., Utah State Univ. 1964. Powder TechnoL,
5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS 3 M. D. Ashton, D. C . H. Cheng, R Farley and F . H. H . Valentin. Some investigations into the strength and flow properties of powders, RheoL Aria, 4 (3) (1965) 2064 J. C. Willia+ns and A- H. r'-ks, The preparation of powder specimens for shear cell testing. RheoL Aria, 4 (3) (1965) 170 . 5 V. V. Sokolowski, Statics of Soil Media (transL by D. H. Jones and A- H . Schofield), Butteraorths, London, 1960. 6 J_ C. Williams and A. H . Birks, The comparison of the failure measurements of powders with theory, Powder TechnoL, 1
(1967) 199.
343
7 A- H. Birks, Lecture delivered during the course on "Failure Properties of Powders" at the University of Bradford, March 19708 J . F. Cam and D. M. Walker, An annular shear cell for granular material, Powder TechnoL, 1 (6) (1968) 3699 M_ D. Ashton, P F . Farley and F. H . H. Valentin, An improved apparatus for measuring the tensile strength of powders .
J. ScL Inm 41 (1964) 763. Effect of particle size upon 10 R Farle_v and F_ H. H. Val_ntin Powder TechnoL, 1 (6) (1968) 344_ the strength of powders,
Powder TeclmoL . 5 (1971!72)
329
The Failure Properties of Lactose and Calcium Carbonate Powders SILVA KOtOVA and N_ PILPEL Department of Pharrracv . Chelsea College of Science_ linirersirr of London- .tfanresa Road London_ S_ 11 _ 3 lGr . Briraini (Received March 12 . 1971 : in revised form Au_eust 26. 1971)
Summary An annular shear cell has been employed to investigate the failure properties of taro cohesive materials, a-lactose monohydrate and calcium carbonate, and to classify then by the way in which their fundamental parameters depend upon their state of consolidation_ It has been shown that the two materials are examples of "simple" powders ; the angle of internal friction, A, provides a satisfactory single parameter for describing their properties. A general relationship between A and the particle size of the powders has been established over the sire ranges and stresses investigated .
INTRODUCTION Jenike r -2 and others` extended the basic principles of soil mechanics to describe the flow and failure properties of fine, cohesive powders, on the assumption that they could be treated as modified Coulomb solids . When such powders are consolidated and sheared under an applied normal stress they fail_ By plotting the shear stress at failure against the applied normal stress (or alternatively, against the compound normal stress), one obtains the powdei s yield locusA separate yield locus is obtained for each state of consolidation or packing density of the powder bed_ The end points of the loci are of particular significance, since they represent conditions where the powder is failing in shear without a change in volume occurring . Various equations have been proposed to describe the shapes of the loci'-' -'. In particular, Williams and Birks 6 showed that when the end points fell on a single curve passing through the origin of the coordinate system employed, it was possible to express the failure properties of the powder by means of an angle of internal friction and other fundamental parameters .
If these parameters are independent of the packing density, the powder is described as "simple". but if one or more of them vary with the packing density, the powder is described as "complex_ Relatively little data have so far been published for distinguishing between the two categories 6_ In the present work . shearing tests were carried out in an annular shear cell', instead of the more usual type of Jenike cell', on two different materials x-lactose e monohydrate and calcium carbonate. over a range of particle sizes and packing densities. The results were combined with those obtained in a tensile tester_ The resulting loci were analysed to see whether their shapes conformed to the equations proposed by Williams and Birks 6 •' and whether the failure properties of the materials could therefore be exprecsed in terms of an angle of internal friction and other fundamental parameters . One purpose of the investigation was to assess the operation and performance of the annular shear cell ris-a-ris that of a Jenike cell_ Other objectives were to establish whether the two materials investigated behaved as "simple" or as "complex" powders and to see whether any correlation could be established between the particle sizes of the powders and one or other of their fundamental parameters .
EXPERIMENTAL Materials
The two materials investigated were x-lactose monohydrate (Whey Products Ltd) and precipitated calcium carbonate (Solvay et Cie and J . & E_ Sturge)_ The former was classified into close size fraction on an Alpine airjet siever or a Bahco classifier, the latter on an Alpine classifier (by courtesy of the Warren Spring Laboratory)The lactose was representative of amorphous, organic powders having relatively low particle Ponder Tec noL 5 (1971,772)
330
S_ KOeOVA. N . PILPEL
TABLE I Physical properties and particle sizes of fractions of lactose and CaC0 3 Calcium carbonate
7-Lactose monohrdrate Size range (ltm)
Average size (pm)
Size range (urn)
Average size (!tm)
3-4 4-7 5.5-8 .5 7.5-9.5 12-0-15-0 40-55 Molecular weight Melting point. ° C Particle density . g/cm 3
4.0 5-5 7.0 8.5 135 47 .5 360.3 210-214 (for range 0-50 pm) 1 .550-1 .552 0.01
0-3 3-5 8-12 12-14 14-16 20-30
1_5 40 100 130 150 250 100.9 1339 2.711
Moisture content . °o w,w
density and melting point ; the calcium carbonate was representative of crystalline inorganic powders with a high particle density and melting point . Relevant physical properties of the two materials are given in Table 1 together with the particle size range and sizes of the various fractions . These were determined by optical microscopy_ The fractions were dried at 105 ° C until their moisture contents were ca. 0.01 % (w/w). This level was maintained throughout by performing the experiments in a room fitted with a Drymatic 50 dehumidifier, which kept the ambient relative humidity at a constant value of 60± 5 % . Apparatus and procedure Two pieces of apparatus were used for investigating the failure properties of the materials . (a) The tensile tester (constructed by K . C_ Products) was of the same design as used by Ashton et al_'_ It consisted of a shallow, cylindrical brass dish. 9 cm i .d. and 1 .0 cm deep and was split vertically in half, one half being fixed to the base, the other resting on steel ball bearings which moved in grooves . The powder was introduced into the cell, the two halves being clamped together, and it was then consolidated by twisting the lid ten times through an angle of 20°, under various normal loads . The movable part of the cell was unclamped and an increasing tension was applied to it through a calibrated spring, connected to a shaft moving at 0.125 cm/min. The tensile stress of the powder and the particular packing density concerned were calculated from the time taken for the two halves of the cell to
001
separate, and from the weight and volume of the powder filling it . By applying different consolidation loads, tensile stress and packing density values were obtained, which were fitted to a best straight line by regression analysis. The equation of this regression line was then used to calculate the tensile stress for any value of the packing density achieved during shearing of the powder in the shear cell . (b) The shear cell (kindly constructed by Glaxo Laboratories Ltd .) was of the annular type' and is shown in Fig. 1(a) and diagrammatically in Fig. 1(b). It consisted of an annular shaped trough covered by an annular lid containing flights at 20° intervals. The flight level defined the shear plane between the powder in the lid and that in the cell . The lid was restrained during shearing by a hardened tempered steel bar equipped with strain gauges (silicon semi-conductor types-2A-1A-120P ( + ) and 2A-1A-120N(-) from Ether Ltd .). Both arms of the bar were restrained to minimise tilting of the lid and ensure accurate recording of the torque being exerted on it. A normal load was applied via a platform attached to the shaft of the lid; it was possible to reduce the weight of the lid assembly by means of a counterpoise weight and a pulley (Fig . I)30-70 g of powder (depending on the particle density) were needed to fill the apparatus (permitting a large number of measurements to be made on relatively small samples) . But because of the design, the shear strain varied across the width (r_-r 1) of the shearing area . This meant that the powder at the outer wall was being sheared more than that at the inner wall . However, errors were minimised by Ponder Technol_ 5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS
making (r 2 -r 1 ) small and by employing a low rate . of shear of I rev/hour The torque developed during shearing was measured from the output of the strain gauges. These formed four arms of a Wheatstone bridge which %vas connected to a pen recorder and chart (Heath Co.) of variable speed and sensitivity. (The connections to the recorder appear on the left in Fig- I(a)_) To prepare a sample for testing the material was introduced into the cell in thin layers, packing it slightly with a light disk- Care -vas taken to ensure uniform height of the bed throughout the cell and thus prevent tilting of the lid. The lid was then lowered on top of the powder and a chosen load Was applied normally . By twisting the lid a predetermined number of times through 20', the powder was preconsolidated to a certain packing density, which could be varied by changing the applied normal load. Although the geometry of the apparatus allowed
331
for unlimited movement during shearing (in contrast to that of the Jcnike apparatus) . the procedure was to minimise the movement by consolidating the sample nearly to its critical state . before shearing it to failure. Consolidation --as completed in the next stageby causing the sample to fail under a given stress until a steady state was reached, or closely approached. This was achieved in the following -vat •_ With the restraining bracket fixed over the steel bar, the motor --as switched on in order to rotate the turn table and the output of the strain gauges was recorded_ The rotation was stopped when the chart reading levelled off (after an initial maximum)_ showing that the sample was then flowing without further increase in the shear stress . Provided that the powder had not become compacted during the consolidation process . the shear force was removed by rotating the lid back by hand, Powder TechnaL 5 (19=1 72)
S. KOCOVA, N. PILPEL
332 the load was reduced and the sample was then again sheared to failure under this reduced load . The normal stress and the shear stress at failure gave one point on the yield locus of the powder at the packing density obtained during preconsolidation of the sampleThe bulk density of the sample was calculated from the volume and weight of powder in the shear cell when full, and the packing density by dividing the bulk density by the particle density of the material- In general a fresh, unsheared sample of powder was employed for the determination of each yield locus. Assuming the shear stress has developed uniformly over the annular area of the shear cell, the total measured torque O_ is given by 8
0 Sym'xi o X
e a 40
Packing density 0336 0380 a414 a455 50
60
Normal stress, 6„ (9/cm 2 ) Fig. 2(a). Yield loci for a-lactose monohydrate. a%crage size 7_0 Pro-
r_
Q =
r,
rr 2 7rdr
Q = 3R -r(r3-ri)
or
(1)
where z is the shear stress, g/cm 2 r) is the inner radius of the cell, cm r, is the outer radius of the cell, cm . By applying known torques to the apparatus (by hanging weights over the system of horizontal and vertical pulleys) and noting the readings on the recorder chart, it was possible to convert the readings into values of shear stress T by means of the above equation (1). Relatively little data are available on the shear behaviour of lactose and CaCO 3 t o and it was not possible to make a direct comparison between the present results and those that would have been obtained in a Jenike cell because of the smallness of the fractionated samples available. However, measurements made in the annular cell on a sample of titanium dioxide gave a value for A within 2° of that obtained' in a Jenike cell . Measurements in an annular cell tend to veer on the side of safety in regard to predictions about critical blocking of apertures for hoppers and bunkers' . Other advantages of the annular cell over the Jenike cell were that it enabled one to employ relatively small amounts of powder and considerably simplify the procedures of consolidating and shearing the samples. RESULTS
The experimental results and the parameters derived from them are given in full in Tables 2 and 3 .
20 30 40 50 Compound normal stress
4+T (g/cm 2 ) Fig. 2(b). Yield loci for a-lactose monohydrate, average size 7.0 pm, using compound normal stress . Figures 2(a) and 3(a) show the yield loci of two representative samples plotted in the usual way with the normal stress aN, g/cm2 as the abscissa and the shear stress at failure, r, g/cm 2 as the ordinate_ The line joining the end points of the loci to the origin is the critical compaction yield locus and its slope is denoted 6'. Figures 2(b) and 3(b) show the corresponding modified yield loci, which were obtained by plotting the shear stress at failure against the compound normal stress (aN +T) or (a3), g(cm2. The line joining the end points of the modified loci to the origin is the internal friction yield locus and its slope, A, is the angle of internal friction of the material for sustained yield . In Figs. 4 and 5 all the yield loci have been reduced to a series of parallel straight lines by plotting log T against log (aN +T) (or log aN)_ It was not possible to obtain sufficient experimental points at any particular packing density to justify fitting each line by regression analysis . But since the original loci at Powder TeehnoL, 5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS
333
TABLE 2 Experimental results and derived parameters . a-Lactose monohvdrate Tensile stress.
Packing
Applied
density,
normal
T
pa/pp
stress_ ~c_N
(g/~')
(gfcm - ) Arerage size
Compound normal
stress. cti (9/c"2 )
r (g: czn9
Reduced
Reduced
normal
compound normal
stress e.,'GE
Reduced shear stress, r'CE
stress.
Reduced
compound shear sties . r'eE
G~'r'E
4.0 Ian
1-05 0-152 265 3 .70 1.05 0.152 4.05 5-10 1-05 0-152 7_20 8.25 1-05 0-152 8-85 9-90 232 0-172 2-65 4-97 232 0-172 6-15 8 .47 232 .172 0 12.25 1457 2.32 0-172 17-42 19-74 3.65 0222 1150 15-15 3-65 0.22 21 .90 18-25 3-65 0.222 26.10 29.75 0-)-)-) 3-65 37-85 3420 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S, deg Angle of internal friction, deg Deviation from reduced stresses curie Deviation from linearised yield loci Average
Shear stress at failure
6.25 6-75 920 11-10 9.2-1 11 .80 17-21 20-00 20-05 2325 30-30 38-60
0300 0-457 0.815 I-000 0.152 0353 0-703 1.000 0336 0535 0-763 1-000
0374 0-705 0.515 0.762 .835 0 1-040 1 .000 1250 0-252 0.530 0 0-677 .429 0.738 0-990 1-000 1 .146 L402 0588 0583 0-681 0.790 0.880 1-130 1-000 p =03722-2-95 P' = 0338=33 0' .124_73% 4 =0 q° = 0.110=6.3 n = 1 .87 _23 6' =520 A = 49.0 7-0 4.3
0.625 .682 0 0-930 1-120 0.467 0.59S .872 0 1-012 0-532 0-615 0.794 1.020
4.25 8.80 10.75 12.70 7.20 9.50 13 .20 16.10 20.80 18 .00 23.60 3020 3755 40.60
0.076 0.523 0687 I .OOC 0-038 0-134 0-309 0-512 1 .000 0271 0-505 0.750 1-000 0506 0-670 0-830 1-000
0.115 0-495 0.560 1-023 0.703 1.250 1 .000 1 .478 0.117 .428 0 0205 0.565 0366 0.785 0-552 0-960 1 .000 1-240 0314 0567 0535 0544 .765 0 0-950 I-000 1-180 0534 0-864 0-688 .985 0 0-840 1-085 Low 1-175 p =0-320=&7% P° =0298_14-0% q =0.060-5-0 °e q° = 0567 _-5.8 01 n =1 .81 _20% a' =50 A = 48 6.0 5.2
0-467 .967 0 I-ISO 1395 0393 0.520 0-722 0 .880 1-136 0 .533 0=7(10 .895 0 1 .110 .815 0 0-930 1-025 1-110
size 5.5 lnn
0.109 050 0.65 1 .15 0-50 0.109 450 5.10 0.50 0.109 5.90 6 .40 050 0-109 8-60 9.10 1 .50 0212 0-65 2.15 1 .50 0212 225 3.75 1 .50 0212 5.20 6.70 1.50 0.212 .60 8 10.10 1.50 0.212 16.80 18.30 0280 &60 10.60 2.00 200 0280 16.00 18-00 200 0280 23.80 25-80 200 0280 31 .75 33 .75 275 0.345 23.80 26.55 0345 275 3150 3425 275 0.345 41.75 39.00 275 0345 47.00 492 :5 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) S hear index Angle S', deg Angle of internal friction, deg 7 Deviation from reduced stresses curve Deviation from linearised yield loci
4625
51 .05 55-25
(conrinued) Powder Tech oL. 5(197 ;/72)
S. KOCOVA, N . PILPEL
334 TABLE 2 (continued) Tensile stress, T (g/cm 2)
Packing density pB/pp
Applied normal stress, cN (g/cm2)
Compound normal stress, ar (9/pn2 )
Shear stress at failure, T
Reduced normal stress, aN/cE
Reduced compound normal stress,
Reduced shear stress, VICE
Reduced compound shear stress, =/GE
aV a;
(g/cm 2 )
Average size 7.0 Etm 0.50 0221 1.75 2.25 0.50 0221 325 3 .75 0.50 0.221 4.95 5.45 0.50 0-221 6.35 6.85 0.50 0221 7.90 8.45 1 .00 0.250 1.75 2.75 .00 1 0250 4.95 595 1 .00 0.250 7.90 6.90 1.00 0.250 10.95 11 .95 1_00 0250 16.20 17.20 0-272 1.75 3_75 2.00 2.00 0272 4.95 6.95 7.90 200 0272 9.90 200 0.272 10.95 1295 2.00 0.272 16.20 18.20 200 0.272 23.45 25.45 200 0.272 31.00 33.00 245 0298 23.45 25.90 245 0.298 31.00 33.45 245 0298 38.75 4120 245 0.298 44.45 4200 245 0 298 4620 48.65 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle b', deg Angle of internal friction, deg % Deviation from reduced stresses curve Deviation from linearised yield loci
330 3 .65 4 80 5.85 7_50 4.80 7.25 9.20 11 .25 15.00 8-20 10.25 13.10 14.50 1800 23.00 2930 3020 35.00 39_50 41 .50 43.50
02 2 2 0.412 0 627 0.805 1.000 0.108 0306 0.488 0.675 1_000 0.056 0.159 0.255 0353 0_523 0.756 1 .000 0310 0 671 0.839 0.910 1 .000
0268 .418 0 0.462 0.443 0.645 0.608 0.810 0.740 L000 0-950 .160 0 0296 0346 0.447 0.400 0.567 0.695 0.695 1 .000 .925 0 0_114 0265 0210 0331 0300 0.423 0393 0.468 0.552 0.581 0.772 0.740 1 .000 0.945 0.655 0-532 0.688 0.757 0.845 0.856 0.914 0.898 1 .000 0.941 p =0_191±26% P'=0.724±22 q =0.066±10.6% q° = 0.057-1-103 n =1_72±3 .0% 6' = 43 A =42 3.0 3.0
0390 0_432 0-568 0.692 .886 0 0279 0-421 .535 0 0_655 0.872 0248 0311 0397 0.440 0.545 0.697 0.890 0.621 0.720 0.812 0.853 0.895
625 8.00 9.45 9.75 10.50 10.75 7_65 9.20 11.70 14.05 15.90 7.60 10.30 1250 14.80 17.50 12.50 16.40
0.179 0337 0.442 0.668 0.843 1.000 .111 0 0.308 0535 0_726 1.000 0-10D 0.280 OA70 .660 0 1.000 0.158 0367
0-291 0_427 0518 0-705 0963 1 .000 0224 0.397 0.595 0 .763 1,000 0224 0378 0.543 0.705 1 .000 0.203 .415 0
0.568 0_727 0.859 Q886 0.955 .978 0 0.435 0.521 0664 0796 .905 0 0386 .523 0 0.635 Q751 0.891 0366 .480 0 (continued)
Average size 8 .5 pm 1.50 150 1.50 1.50 1.50 1.50 225 225 225 225 225 270 2.70 270 2.70 270 4.20 420
0384 0384 0384 0384 0.384 0384 0392 0392 0392 0392 0392 0_416 0.416 0.416 0.416 0_416 0.429 0.429
1.70 320 4.20 6.25 8.00 9.50 1_70 4.75 825 11.20 15.40 1.70 4.75 8.00 11.20 17,00 4.75 11 .00
320 4_70 5.70 7_75 9.50 11_50 395 7.00 1050 13.45 17.65 4.40 7.40 10.70 13.90 19.70 895 1420
0_658 0.842 0995 1.026 1.105 1_132 0-497 0.598 0.760 0915 1.030 0-447 0.605 0.735 0-871 1.030 0.417 0547
Powder TechnoL,5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS
335
TABLE 2 (continued) Tensile stress_ T (g/cm2)
Packing density Pe/PP
Applied normal stress,
Compound normal stress,
Shear stress at
Reduced normal
failure,
stress,
Reduced compound normal
GN
GH
7
Gy/GE
stress,
(q/cm2)
(9/cm -)
(g/cm2)
420 0.429 15.65 19-85 420 0.429 30-00 3420 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S', deg Angle of internal friction, deg Deviation from reduced stresses curve Deviation from linearised yield loci
z;GE
Reduced compound shear stress, T., GE
fir/GE
2030 30 .75
0.522 1-000
3.87 6-28 7-61 4.45 6-05 850 10-21 11-73 13-10 635 820 1028 11-87 14-58 2032 2250 24-41 1321 1435 17.84 27.68 3832 49-61
0211 0.809 1.000 0.109 0301 0-417 0-686 0.885 1 .000 0.056 0-156 0268 0355 0517 0-755 0.915 1.000 0.078 0.133 .258 0 .500 0 0.746 1 .000
7
Arerage size 135
Reduced shear stress,
0-580 0-676 1 .000 1-025 p = 0-39517-6 °e p- = 0334-10-6 %e q =0-351 x-73 %e q° =01324-3 .6%. n = 155 -L4-0% X = 445 A = 41.5 8-0 5-7
0-594 0.900
gm
0-19 0250 1-75 1.94 0-19 0250 6-72 6.91 .19 0 0250 831 850 0.35 0.274 1.75 2.10 035 0274 4.85 520 035 0274 6.72 7-07 035 0274 11.05 11 .40 035 0-274 1422 14-57 035 0274 16.10 16-45 0-68 0282 1 .75 2.43 0.68 0282 4.85 5.53 .68 0 0282 831 899 0.68 0.282 11 .05 11 .73 0.68 0282 16.10 16.78 0.68 0282 23.50 24.18 0.68 0.282 28.48 29.16 0-68 0282 31 .17 31 .85 1 .40 0350 4.85 6-25 1-40 0350 830 9.70 1.40 0350 16.10 17.50 1.40 0350 31 .17 3257 1.40 0350 4650 47-90 1 .40 0350 6235 63.75 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile su~ (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S, deg Angle cf internal friction, deg Deviation from reduced stresses curve %Deviation from linearised yield loci
different packing densities were geometrically similar in shape, it was considered legitimate to derive a common slope from all the results, i.e. 1/n,- where n is the shear index of the material, and then fit
0-228 0-466 0.814 0-755 1-000 0915 0271 0-128 Q316 0368 0-430 0-528 .693 0 0-635 0-886 0-729 1 .000 0 .813 0-076 0204 0-174 0264 0_.82 0331 0-369 0.382 0-527 0.4f-, 0.760 0-652 .72 0-915 0 1-000 0-785 0-098 0.212 .152 0 0-230 0 0.274 .286 0510 0.445 0-752 0-615 1-000 .795 0 P =0-148149 p" = 0-145-2-5% q =0-022=45 q~ =0-022-4-0% n = 1_.8 =6 .0 S' = 383 A = 3825 30 9.8
0-355 0-739 0-895 0-271 0368 0-517 0b2D 0-714 0.795 .199 0 .257 0 032 2 0373 0-458 0-638 0-706 0-768 0.2 208 0-)-) 5 0280 0.435 ObO5 0-778
individual lines, with this slope, through the experimental points appropriate to each packing density. The stan dard deviation of the points about these lines was calculated and the figures given at the end Powder Tecimoh 5 (1971172)
S. KOCOVA, N. PILPEL
336 TABLE 3 Experimental results and derived parameters . Calcium carbonate Tensile stress. T (g/cm2)
Packing density, ps lpp
Applied normal stress,
Compound normal stress,
a-%
c,
(g/cm2 )
(9/cm 2 )
Reduced applied normal stress,
Reduced compound normal stress,
e-dal.
e.%/ee
1 .80 525 830 4 .90 9.01 11-15 1428 1732 18 .02 10.71 13-05 17-50 25.75 15.20 23-98 28.81 33 .01 34-05 19-23 3272 5250 59.05 59.50
0-204 0-627 1 .000 0.125 0309 0-444 0-694 0.875 1 .000 0.216 0357 0-559 1 .000 0-267 0.514 0.710 0.949 1 .000 0-147 0-570 0-850 0 963 1 .000
0232 0.238 0-648 0.677 1 .000 1-071 0.164 0303 0340 0.557 0512 0.689 0-708 0.883 0-880 1 .070 1 .000 1 .113 0.250 0-462 0385 0563 0578 0-755 1 .000 1 .111 0300 0.490 0536 0-772 0.723 0-928 0 .952 1-063 1 .000 1 .097 0.183 0353 0.589 0.601 0.964 0-957 0.964 1 .084 1 .093 1 .000 p =0.23±9.4% p" = 0214±9-8 q = 0.046±5-0 a" = 0.044 x-4.5 n =1 .48±25 0//. S' = 48.0 A = 47.5 3 .4 3 .27
0Z 0.648 1 .025 0.289 0-532 0.659 0.843 1 .023 1 .064 0.442 0539 0-722 1 .063 0.467 0.737 0-885 1 .014 1 .046 0.338 0-575 0.923 1-038 1-046
1265 13 .70 17.43 18 .05 13-73 18 .03 22.00 23.58 24-68 15-21 21 .50 33 .05 35.45 35 .65 39.73
0.515 0.624 0.821 1 .000 0341 0540 0.795 0.924 1 .000 0.253 .492 0 0.839 0.931 1 .000 0.404
0.523 0.671 0.822 1 .000 0352 0.576 0-799 0.925 1 .000 0.265 0.500 0.841 0.934 1 .000 .415 0
0.792 0-860 1_090 1 .130 0570 .749 0 0911 0976 1 .021 0-469 .661 0 1 .018 1-091 1.095 .731 0 (continued)
Shear stress at failure r (g/em')
Reduced shear stress, r/cE
Reduced compound shear stress, z/c;
Arcrage size 1 .5 pm 035 0.235 1 .58 1 .93 4.73 035 0.235 5.08 7.75 8 .10 0.35 0.235 0.75 0-245 203 2.78 5.00 5.75 0.75 0.245 7.91 0 75 0.245 8-66 0.75 0.245 1123 11 .98 0 .75 0 .245 14 .15 14.90 0.75 0.245 16 .18 15 .93 1 .05 0-257 5.00 6 05 1-05 0.257 8 .28 9 .33 1 .05 12 .95 14 .00 0.257 1 .05 0-257 23 .28 24-23 1 .50 828 0.274 9.78 1 .50 0274 15.95 17 .45 1 .50 0.274 2103 23 .53 1 .50 0274 29.48 30-98 0274 150 31 .05 32.55 0350 37.98 10_43 2.45 245 0350 31-05 3350 2-45 0350 46.30 48.75 245 0350 5241 54 .86 2.45 0350 54-45 56-90 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle 8', deg Angle of internal friction, deg Deviation from reduced stresses curse Deviation from linearised yield loci
Arerage size 4.0 lire 0.25 .25 0 0-25 025 0.38 0.38 0.38 0.33 038 052 052 052 052 0.52 0.71
0-178 0.178 0-178 0.178 0236 0.236 0.236 0.236 0.236 0351 0351 0351 0351 0351 0.419
8 .10 9.28 12 .91 15.74 8-10 1281 18.90 21 .95 23.78 8 .10 15.74 26.83 29.78 3200 15 .74
8.35 1007 13.16 15.99 8-48 13.19 19-28 2233 24.16 8 .62 16.26 2735 30.30 32.52 16 .45
0.805 0-871 1 .110 1 .145 0.580 0 .757 0.925 0.992 1 .037 0.476 0.671 1 .033 1 .110 1 .113 .745 0
Powder Teehnol, 5 (1971/72)
337
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS TABLE
3
(continued)
Tensile stress, T (4/c"12 ) 0.71 0-71
Packing density, pa/p,,
Applied normal mess, UN
(g/cm2) 0.419 0.419
37.26 39.02
Compound normal mess, a;
Shear stress at failure t,, ~
Reduced applied normal stress,
Reduced compound normal stress,
Reduced shear mess, TIC,
(4/CM2)
(&m2 )
G\IGE
GS/GE
37_97 39.73
41 .03 41 .20
0955 . 1 .000
0_974 1-050 1-000 1-052 p =0247±8 .%o Pr =0243-8 .2%0 q =0-0167--+-8-9% q' = .0163 0 -r 92 ie n = 1 .84 _ 2 .9 ;{ v' = 47.0 A = 46.4
Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S, deg Angle of internal friction, deg Deviation from reduced stresses curve Deviation from linearised yield loci
Reduced compound shear stress. TJeE 1 .030 1 .035
4.0 27
Average size 10_0 fun 035 0.25 0.25 .25 0 040 0.40 0.40 .40 0 055 055 055 0_55 0_55
0292 0-"9 0292 0292 0324 0324 0324 0324 0363 0.363 0363 0363 0363
4.95 732 14.05 1521 7.82 15.25 2132 2355 7.82 15 .48 23 .05 2950 31_05
5-20 8_07 1430 15.46 820 15_65 21 .72 23.95 837 16 .03 23 .60 30-05 31_60
4-23 735 1202 12.52 7.80 13_81 18.95 2020 9.25 1520 20-00 26.70 2750
0326 0514 0.924 1 .000 0333 .649 0 0 .905 1 .000 0252 0-499 0.742 .950 0 1_000
Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S', deg Angle of internal friction, deg Deviation from reduced stresses curve Deviation from linearised yield loci Average size 015 . 0.15 0.15 025 0.25 025 030 030 030 030 030
13.0
0336 0-278 0.477 0.52 7 .925 0 .790 0 0323 1-000 0342 0332 0 .588 0 .653 0.92 7 .805 0 1 .000 0_858 0265 0,298 0507 (1490 .644 0 0.747 0-951 0350 1_000 _870 p =0_083-6 .0% P7=0-082-z61 %e q =0 .01715.9% q"=0-0167 -14-2% n =1 .11=3 .7/.
0-2.74 0.469 0.777 0.810 0326 0377 0.791 .843 0 0.2293 0-481 0-633 .845 0 0.854
3' = 44-0 A = 433 2.6 29
fun
0174 . 0.174 0_174 .191 0 0.191 .191 0 0.211 0.211 0211 0211 0211
7.84 14.05 15.72 7.85 21 .42 2522 735 15.72 2322 29.65 31 .18
8_00 1420 15_87 8 .10 21 .67 23 .47 8 .15 16_02 2352 29.95 31 .48
Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile stress (applied normal stress) Specific tensile stress (compound normal stress)
9.15 1320
0.499 0394
14.00
1.000
9.75 17.65 18.00 1030 1525 2132 26 .00 2625
0338 .922 0 1 .000 0252 .504 0 0-745 0.950 1 .000
0-582 0.504 0.895 0340 1 .000 0-891 0-345 0_420 .923 0 .760 0 1 .000 .775 0 0259 0330 .509 0 .489 0 .747 0 .684 0 .951 0 0834 1 .000 0.842 p =0.133-3A% .132=3.0%'o P- =0 q =a10-=9-8%
0.577 .832 0 0282 0-415 0_752 0-767 0327 Q484 Q677 0226 .834 0
q" =0.009=8.1 %o
(contin ud)
Powder iechnol, 5
(1971172)
338
S. KOl° OVA, N. PILPEL
TABLE 3 (continued)
Tensile stress, T
Packing density, Ps/Pr
(9/tent)
Applied normal
Compound normal
Shear
stress, ar. (gfc n) '
stress,
failure
Reduced applied normal
Cr
=
stress,
compound normal stress,
(g/cm2)
(9/c"12 )
CN/CE
Cr/CE
stress at
Shear index Angle S', deg Angle of internal friction, deg Deviation from reduced stresses cure Deviation from linearised yield loci
Reduced
Reduced shear stress, T/cE
Reduced compound shear stress,
T/cE
n =1.35-25% 6' = 39.8 A = 39.5 3.0 3_1
Arerage size 15 .0 pan
0_15 0363 325 3.40 0_15 6 .05 0363 6.20 0.15 0.363 7_90 8_05 0.42 0_413 325 3_67 0.42 0_413 7.90 832 0_42 0_413 14.75 15_17 0.42 0.413 16.32 16.74 0_42 0.413 =42 2200 0.42 0_413 23_72 24.14 0_75 16 .32 0.480 17.07 0.75 0.480 23.72 24.47 0_480 0.75 31-78 32-53 0.75 0.480 37.95 38.60 0.75 0_480 39.75 40.50 0_95 0.706 23.72 24.67 0.95 0.706 31 .78 3273 0.95 0.706 39.75 40_70 0.95 0.706 4730 48.25 Specific cohesion (applied normal stress) Specific cohesion (compound normal stress) Specific tensile strss (applied normal stress) Specific tensile stress (compound normal stress) Shear index Angle S', deg Angle of internal friction, deg Deviation from reduced stresses curve Deviation from linearised yield loci
1.92 4.05 4.75 2.95 6.55 9_50 11.25 13_76 14.45 14 .50 18.52 21.05 25.00 25.55 19.82 2'-)65 27.75 31.05
of Tables 2 and 3 are the maximum percentage deviations about the mean values. In Figs. 6 and 7 all the points from the yield loci have been replotted on axes of reduced shear stress TR versus reduced applied normal stress c R (where rR = T/cE and cR = cN/cE and cE is the applied normal stress at the end point of the appropriate yield locus). It is seen that for each material, the points fall very reasonably on a single curve (fitted by the method of non-linear correlation) ; the percentage deviation of the points is given in Tables 2 and 3 . Figures 8 and 9 show the straight lines to which the yield surfaces are reduced when the logarithm of the reduced compound shear stress (TR=T/(E E ± T)
0.411 0.766 1.000 0.137 0333 0_622 0_688 0_927 1.000 0_411 7597 0.799 0.955 1 .000 0.501 0.672 0.840 1 .000
0.422 0-243 0.770 0.513 1 .000 0.601 0.152 0.124 0345 0.276 0401 0 628 0.693 0.474 0.929 0_580 1_000 0_609 0_421 0365 0_604 0.466 0.803 0.530 0.953 0.629 1 .000 0.643 0.511 0.419 0_678 0_479 0.844 0.587 1_000 0656 p =0.118±20% p'=0.116±1 .7%
0239 0.503 0590 0.122 0.271 0.394 0.466 0_570 0.599 0358 0_457 0.520 0.617 0.631 0_411 0.469 0.575 0_644
q =0.019±53% q'=0.0183±7.6% n = 1-23-+-3.0% 6' = 31 .0 A = 31 .0 8_0 4.6
is plotted against the logarithm of the reduced compound normal stress (cR=(cr,-i-T)/(cud-T)) . In Figs. 10 and 11 the values of tan A for the different fractions of lactose and calcium carbonate have been plotted against their average particle size in order to establish a relationship between the angles of internal friction of the materials and their particle size. DISCUSSION
It is evident from Figs . 2(a), 3(a), 4 and 5 and also from the results in Tables 2 and 3 that the yield loci Powder TechnoL, 5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS
339
100
15200
S . I.® N
Pocking density Symbol o 0235 x 0.245 0257 a 0 0274 148` 0 0350 1 , I I ∎ I -10 10 20 30 40 50 60 Normal stress, 6. (g/cm° ) 20
1 v 50
P 20
o x c 0
a N 05
Fig 3(a)_ Yield loci for calcium carbonate, average size 1_5 µm .
02 02
+
05
0336 0380 OA14 0455
, 1 I + I 1 1 1 1 10 20 5A IL L 200 500 100 200 500 Compound normal stress, 6u+T(WWcmt)
Fig 4. Logarithm of shear stress at failure as logarithm of compound normal stress for x-lactose monoh3drat- average size 7.0 {tm_
10ao -
soo60_04aoM
, 6„+T (g/cm5 Yield loci for calcium carbonate, average size 1 .5 lam, Fig. 3(b). using compound normal stress 10
20 30 40 50 Compound normal stress,
60
E - 20n rn 1^ h 740L n 60. S 4oL N
of all the samples investigates satisfactorily obey the Warren Spring equation 3 r ° 6N +T (2) CC
T
This equation employs the compound normal stress (6N + T) (as illustrated in Figs_ 2(b) and 3(b) instead of 0N, for deriving information about the failure properties of a powder- The advantage of using this modified yield locus is that it allows for the tensile stress of the powder, as a component of the total normal stress during shearingWilliams and Birks6 •' concluded from theoretical considerations that it is only at the end point of a modified yield locus (designated by the subscript E) that all the work being done by the shearing force is being spent in overcoming the forces of internal friction within the powder. At all other points work is being done against both internal and external
forces, due to changes in the volume of the powder
Symbol x A d
2.0-
.
Packing density 0245 0257 0274 0350
.0 .0 1000 2.0 40 60 10 ZOO Ono 60 Compound normal stress, 64 = w~ 1 (g/~nz) Fig_ 5. Logarithm of shear stress at failure rs logarithm of compound normal stress for calcium carbonate, average size 15 fun 1.0
bed_ They therefore selected this end point as more characteristic of a powder than any other point on the locusEquation (2) does not give any information about the end point of a yield locus or predict that an end point should occur- This is contrary to what has been shown by Jeniker_ For the end points, eqn_ (2) can be rewritten C° _ zE (3) T a; and in cases, as in the present work, Figs 2(b) and 3(b), where the end points of the modified loci lie Powder Technol., 5 (1971/72)
S. KOCOVA, N- PILPEL 1.0
0
Symbol o x A O L
-02
Packing density 0336 0.360 Q414 0 .455
02 .4 0 O5 0.8 Reduced applied normal stress
1
Fig_ 6 . Yield loci for a-lactose monohydrate, average size 7A fan, using reduced stresses .
0]
0.2 0.4 06 08 to Reduced compound stress, (6,•+ T)/(?E+T)
Fig. 8. Linearisation of family of yield hydrate, average size 7.0 fun.
loci for a-lactose
mono-
2.Or
QiO/
10 0.6
VW F-
()S
N H
a
L /1.~w
07
N V
02
O
rN 0.3
t , a 0
02
7
Symbol o x a 0
b d v u 02
0.4
OS OS Reduced applied normal stress, V(SE
•
Fig 7. Yield loci for calcium carbonate, average size 1 .5 fun, using reduced stresses. W .1 0
on a single line through the origin (the internal friction yield locus), one can also write : TE = tan A(C + T) = tan AaE (4) where A is the angle of internal friction of the powder for sustained yield and is a characteristic property of it . Combination of eqns. (2), (3) and (4) then gives' : T = tan A UN(11n) (TE(x-1)/n (5) In view of the unique properties of the end points of the yield loci, it is advantageous to use them in order to calculate the so-called reduced stresses of powders undergoing shear . These are dimensionless
Packing density 0.235 0.245 0.257 0.274 0.350
, Q4 0.6 O.B to Reduced compound stress, O, rtC tT)f(6E +T) 02
2.0
Fig 9. Linearisation of family of yield loci for calcium carbonate, average size 1 .5 fan.
E
quantities' and can be substituted into eqn . (5) to give
1j"
r =tan A cR (6) Here TR is the reduced compound shear stress (as defined in the List of Symbols) and cR is the reduced compound normal stress. By inspecting families of yield loci at different packing densities, Williams and Birks came to the conclusion that for two materials at least, Ti0 2 Powder Tecin+oh 5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS
that have been employed . In these circumstances, one can further simplify the presentation of the yield loci (Le- linearise the yield surfaces) by expressing eqn. (6) in logarithmic form
tz0-
v O
X
O ;, 1D0 xx
~090 I 0
c
\x
x
a
a 0so 05 0 • 5.0 10.0 15.0 20 .0 25 .00 Particle size, t. (jam)
Fig. 10- The effect of particle for a-lactose monohydrate.
•
341
5.0
tao
1s0
Particle
size on
20.0
30.0
500
the angle of internal friction
• 250
30 .0
size Dp (pm) Fig. 11 . The effect of particle size on the angle of internal friction for calcium carbonate.
and penicillin, the cohesion and the tensile stress were both linear functions of a E and aE. This enabled them to define two quantities q (alternatively q") and p (alternatively p"), called respectively the specific tensile strength and the specific cohesion. The same conclusion is now arrived at concerning the samples of lactose and calcium carbonate that have been investigated in the present work- This is shown by the relative constancy (despite a certain scatter) of the values of p, p", q and q" listed in Tables 2 and 3 and also in Figs . 6 and 7, where the reduced loci intersect the ordinate at p and the abscissa at q. The deviation of the points from these loci is about 3-4% (not more than 10% in the worst cases, see Tables 2 and 3), which is considered very satisfactory, bearing in mind the nature of the correlations
log rR = log tan A i I log cg (7) n A straight line of slope 1/n should be obtained by plotting log rR versus log 7R. This is confirmed for both lactose and calcium carbonate in Figs 8 and 9_ Powders for which the quantities A, q", p" and n are independent of the packing density may be termed "simple powders", while those for which they vary with packing density are called "complex"'On this definition, titanium dioxide, pencillin, lactose and calcium carbonate are all examples of simple powders over the size ranges and stresses that have been investigated_ The determination of A, in which c5. is replaced by (cr ~-T), involves a complete analysis of all the stresses that are acting in the plane of shear of the powder. Unlike other quantities, such as C and T, A is independent of the packing density of the powder_ Although the quantity n is also independent of the packing density, A is the only parameter that relates to failure of the powder under steady state conditions, that is, at the end point of the yield locus, where no change in volume is occurring . (in passing, we consider it important that the locus should be fitted to pass through the point T_ Though its determination involves experiments of a different type to those in a shear cell, with the possibility of producing errors in the values of n, the only way of independently confirming the validity of the T measurement would be to obtain other values of -r at negative values of a. Such measurements are not, at present, possible_) A is therefore likely to be a useful parameter for assessing the failure properties of powders. It is also likely to be useful in practical matters, such as the design of hoppers, calculation of critical blocking apertures, etc_ where steady state conditions are obtained during flow. If; therefore, it is concluded that A is a satisfactory single parameter for assessing the failure properties of simple powders, it is of interest to see not only how A varies from one powder to another but also, for example, how it varies with the particle size of the powderIn Figs . 10 and 11 the values of tan A for the different fractions of lactose and calcium carbonate Powder Tecimol . 5 (1971x72)
S. KOCOVA, N. PILPEL
342 have been plotted against their average particle sizes- The two curves are seen to be geometrically similar. Within the range of sizes examined, they have the general form tan A = K+ S exp (-*Dp) (8) where DP is the average particle size of the fraction and K, S and yi are constants . These constants are evaluated by subjecting the curves to treatment on an analogue computer. It is found that for lactose (4Eunn < D P < 50 pm) tan A = 0.77+0.47 exp (-0.019 Dp) (9) and for calcium carbonate (1 .5 . un < DP < 30 ,um) (10) tan A=0 .46+0.64exp(-0.0031Dp) It seems probable that the general equation (8) will also be found to apply to other powders over particular ranges of their particle size provided that they conform to the definition of "simple" powders in which A is independent of the packing density of the powder. CONCLUSIONS
(1) The annular shear cell gives reproducible results for determining the yield loci of powders over a range of particle sizes and packing densities . Measurements can be made on relatively small samples of powder, more rapidly and conveniently than in a Jenike cell . (2) The failure properties of lactose and calcium carbonate have been expressed in terms of four fundamental parameters. These are the angle of internal friction A, the specific tensile stress q", the specific cohesion p" and the shear index n . Since A, q", p" and n have been shown to be independent of the powders' packing densities over the size ranges and stresses investigated, the two materials are examples of "simple" powders . (3) A provides a convenient parameter for specifying the failure properties of "simple" powders . Relationships have been established between A and the average particle sizes of the lactose and calcium carbonate, which are of the form tan A = K + s exp(-%Dp).
sion for helpful discussions and the Directors of I.C.I . for financial support. We also thank Glaxo Laboratories and the Warren Spring Laboratory for the gift of apparatus and materials . LIST OF SYMBOLS
C=C(pB) cohesion, g/cm 2 DP representative particle d iameter. pm n shear index, the exponent of the powder's yield locus equation specific cohesion of the powder (applied P=CICE normal stress) p =` C/ci specific cohesion of the powder (compound normal stress) q=T/cE specific tensile stress (applied normal stress) q" = TIC; specific tensile stress (compound normal stress) rl inner radius of shear cell, cm r2 outer radius of shear cell, cm T=T(p B) tensile stress, g/cm 2 b' angie of inclination of the locus joining the end points of family of yield loci when applied normal stress is used A angle of internal friction of the powder constants K, bulk density, gfcm 3 PB particle density, gcm 3 PP applied normal stress corresponding CE to end point of powder's yield locus . cE=cE +T compound normal stress at end point of yield locus applied normal stress, g/cm 2 UN cr=cN +Tcompound normal stress, g/cm 2 aR=CN/CE reduced applied normal stress cR=cr/aj reduced compound normal stress T shear stress at failure, g/cm 2 shear stress at end point of yield locus TE reduced shear stress (applied normal TR = T/GE stress) reduced shear stress (compound normal TR=TIcE stress)
REFERENCES ACKNOWLEDGEMENT
We wish to thank I .C.I. Pharmaceuticals Divi-
I A. W. Jenike, BuIL 108, Eng. ExpL Sta., Utah State Univ ., 1961. 2 A. W. Jenike, Bull. 123, Eng Expt . Sta., Utah State Univ. 1964. Powder TechnoL,
5 (1971/72)
FAILURE PROPERTIES OF LACTOSE AND CALCIUM CARBONATE POWDERS 3 M. D. Ashton, D. C . H. Cheng, R Farley and F . H. H . Valentin. Some investigations into the strength and flow properties of powders, RheoL Aria, 4 (3) (1965) 2064 J. C. Willia+ns and A- H. r'-ks, The preparation of powder specimens for shear cell testing. RheoL Aria, 4 (3) (1965) 170 . 5 V. V. Sokolowski, Statics of Soil Media (transL by D. H. Jones and A- H . Schofield), Butteraorths, London, 1960. 6 J_ C. Williams and A. H . Birks, The comparison of the failure measurements of powders with theory, Powder TechnoL, 1
(1967) 199.
343
7 A- H. Birks, Lecture delivered during the course on "Failure Properties of Powders" at the University of Bradford, March 19708 J . F. Cam and D. M. Walker, An annular shear cell for granular material, Powder TechnoL, 1 (6) (1968) 3699 M_ D. Ashton, P F . Farley and F. H . H. Valentin, An improved apparatus for measuring the tensile strength of powders .
J. ScL Inm 41 (1964) 763. Effect of particle size upon 10 R Farle_v and F_ H. H. Val_ntin Powder TechnoL, 1 (6) (1968) 344_ the strength of powders,
Powder TeclmoL . 5 (1971!72)